NUCLEAR MAGNETICRELAXATION IN A SPIN 1

SYSTEM

K.R.Jeffrey

Department of PhysicsUniversity of Guelph

Guelph, Ontario, Canada

[.INTRODUCTION

Deuterium nuclear magnetic resonance (2HNMR)has become a very valuable tool in the study of thestructure and dynamics of anisotropic fluids,especially lipid bilayer model and biological mem-branes (1,2). Because the deuterium nucleus hasspin / = 1, there is a quadrupolar splitting of thenuclear Zeeman resonance line into a doubletwhenever the nucleus experiences a non-zero elec-tric field gradient. In anisotropic fluids, thequadrupolar splitting for deuterons, which havebeen substituted for hydrogen atoms bonded to car-bon atoms, gives a direct measure of the orienta-tional order parameter for the C - D bond. Thisparameter can be related to fundamental structuraland dynamical properties of these systems (1). While2HNMR spectroscopy is a well established techni-que, the use of the nuclear magnetic relaxationtimes to study the dynamics of anisotropic fluids isjust beginning (3,4).

For a system of noninteracting spins, /= Vz, thereare two relaxation times, commonly called the spin-spin and spin-lattice relaxation times T2 and Turespectively.

For the case of spin /= 1 nuclei, where there is anon-vanishing electric quadrupole interaction, thereare 5 independent relaxation times to be measured(5). One of these is associated with the return of theZeeman energy to its equilibrium value and istherefore called T,z, another is associated with thedecay of the quadrupolar energy, 7",Q, and there arethree independent spin-spin relaxation times.Jacobsen and Schaumberg (6), using adiabatic rapidpassage techniques, measured Tlz and TiQ separate-ly for the deuterium quadrupolar doublet of D2O in anoriented lyotropic liquid crystal system. Void andVoid (7) reanalysed the relaxation behaviour for aspin 1 system using the Redfield formalism (8) andshowed how TIZ, T1Q, and one spin-spin relaxationtime, 7"2, could be measured using selective and non-selective pulse techniques. T2 is the relaxation timewhich can be obtained from a measurement of thelinewidth in the absence of temperature, field, andother macroscopic inhomogeneities. T2 alsodescribes the decay of the quadrupolar echo (9) as afunction of the spacing between the two radio fre-quency (RF) pulses used to form the echo. Thetechniques of Void and Void are applicable to thevery narrow line spectra found in thermotropic liquid

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Vol. 3, No. 2-4 69

crystals. However, the linewidths in lyotropic liquidcrystals, especially model membranes, preclude theuse of selective pulses. In many circumstances,unoriented or powder samples must be used.

Ahmad and Packer (10,11), using the Redfield for-malism, have considered the response of a spin 1system to two three-pulse sequences when thereare both secular and time dependent quadrupolar in-teractions. The two sequences are denoted by

and(n/2),-T-(n/4),- r-(n/4),

(n/2)x-T-(n/2)y- T-(nl2)y

where the symbol (6)k indicates an RF pulse appliedalong the k axis for a time 6 = yHJ. W, is the strengthof the RF magnetic field. The (n/2), - T - (n/4)y - T-(n/4)y sequence, commonly called the Jeener-Broekaert sequence (12), is of particular importancebecause it provides a means of measuring T1Q. Thissequence and the conventional (n), - r - (n/2), se-quence were used to measure TlQ and T,z for thedeuterium quadrupole doublet of D2O in an orientedlyotropic liquid crystal system (11).

Vega and Pines (13) have developed an operatorformalism in terms of nine fictitious spin Vz operatorsin order to present a very clear picture of the evolu-tion of a spin 1 system in the presence and absenceof RF fields. However, they did not consider the ef-fects of relaxation.

In the present discussion, the operator formalismof Vega and Pines (13) is used to develop a numberof pulse techniques suitable for measurement of therelaxation times for a spin 1 system. While some ofthese have already appeared in the literature, it isworthwhile presenting a complete picture using oneformalism. In addition, the various techniques will beillustrated using two anisotropic fluid samples. Thefirst is a mixture of 30% potassium laurate-ad2, 60%H2O, 6% decanol, and 4% potassium chloride. Thissample is a nematic lyotropic liquid crystal (14). Thespectrum of the deuterons in the a position in thepotassium laurate molecule is a single doublet pat-tern characteristic of a spin 1 system in an orientedsample. There is a single angle equal to 90° betweenthe effective axis of symmetry of the electric fieldgradient and the magnetic field. The other sample isa mixture of 70% rubidium stearate-ad2 and 30% H2Oand is also a lyotropic liquid crystal but the stearatemolecules do not orient in the magnetic field. Thedeuterium spectrum is a typical powder pattern, in-

dicating that the angle between the axis of symmetryof the electric field gradient and the magnetic fieldmay take all possible values.

II. MEASUREMENT OF RELAXATIONTIMES IN A SPIN 1 SYSTEM

A Relaxation Times for a Spin 1 System

The Hamiltonian for a system of non-interactingspins, /= 1, in an external magnetic field with a non-vanishing electric quadrupole interaction can bewritten as

ac=3cz -where

3 C Z = h(x>0<

is the Zeeman interaction and

D]

(D

(2)

(3)

is the secular or static part of the electric quadrupoleinteraction. The Zeeman interaction strength inangular units is

while the quadrupolar interaction strength is

ojQ = AQ[(3cos2e-1)/2 + r)Sin20cos2if] (4)

where= 3{e2qfO/(2/-1)}/2. (5)

The orientation of the principal axes of thequadrupole interaction tensor with respect to the ex-ternal magnetic field is given by the angles 6 and ip,and r] is the asymmetry parameter. It will be assumedthat COQ « coo and that only the first order perturba-tion of the quadrupolar interaction on the ZeemanHamiltonian need be considered. The Zeeman andstatic quadrupolar interactions determine theobserved deuteron spectrum.

The interaction of the spin with RF radiation isdescribed by

i lxcos(wt+ (6)

where 2ftw, is the RF irradiation strength at frequen-cy a;. The relaxation of the spin system back toequilibrium is determined by the time dependentpart of the quadrupolar interaction

Kr{t)=XQ-<XQ(t)>-

70 Bulletin of Magnetic Resonance

The relaxation Hamiltonian may be written

3Cr(O=eO/[6/(2/-1)]

\- U)-4r,P] (7)ii'

where (y,/3 = x,y,z) and the VyP are the components ofthe electric field gradient tensor in the laboratoryframe.

The dynamics of the spin system are best dealtwith in terms of the density matrix. The timedependence of the density matrix is described by

dQ(t)ldt=ilh[X(t) (8)

In order to eliminate the effect of the large Zeemaninteraction, Eq. 8 is transformed into the rotatingframe. Using the unitary operator

S=exp(-kotl.) (9)

the equation of motion becomes

deldt= ilh[Q, V+Kr(t)] (10)

where Q = SgS1

V=h&wlz + V 3 / 7 w 0 ( 3 / , 2 - I2) + hw,lx,

a n d

3CAt)=S5CAt)S'.

Eq. 10 still has both time-dependent and time-independent parts of the Hamiltonian in the com-mutator. In order to remove the time-independentpart, a further transformation to an interactionrepresentation is performed. The transformation isdefined by

r = e x p ( - / A c o t f , + V z i c o Q t ( 3 l z2 - I2) + ( 1 1 )

The equation of motion in this representation is

dQ*ldt=ilh[Q*,Kr*(t)] (12)

where

and

Provided that the relaxation is sufficiently weak,the effects of K r(f) can be evaluated using time-dependent perturbation theory (14) which leads tothe approximate equation

(13)

where the bar indicates an ensemble average andQ*,g is the equilibrium density matrix.

To obtain the equation of motion for Q, the densitymatrix in the rotating frame, it is necessary totransform back into that frame.

d e l d t = l l h [ Q , V ] -

(14)

If the molecular motion which gives rise to the timedependence of3Cr is characterized by a correlationtimer,, then Eq. 13 is valid if [|3Cr(OI2T,2]"2 is a smallnumber.

Following the approach of Hoffman (15), the densi-ty matrix is expanded in terms of operators formedfrom the spin angular momentum operators. For aspin 1 system, the density matrix may be expandedin terms of the unit operator 0, and eight other in-dependent operators Q2...Qe. Vega and Pines (13)have used nine symmetrized operators, of whicheight are independent, and the unit operator. Thenine operators /„,, where p = x,y,z and i = 1,2,3, aregiven by

/,, = VH (15)

(16)

(17)

The symbols p,q,r are cyclic permutations of x,y,z. Itshould be noted that

L (18)

Wokaun and Ernst (16) have chosen to use a set ofdifferent linear combinations of these operatorswhile Jacobsen et al (5) have chosen to use the ir-reducible tensor operators of first and second rankderived from the spin angular momentum operators.

For this discussion, it is convenient to define a setof operators Q, (j = 1-9) where Q, is the unit operator,O2.. Q7 are the six operators defined by Eqs. 15 and16, and

Qt=l,i = V2(I2-I2)

Q,= L- ly3 = V2(3L>- F).

The density matrix may be expanded as

(19)

(20)

(21)

The time dependence is contained solely within thecoefficients Aj(t).

The observed signals in an NMR apparatus areproportional to tr(g/,) and tr(e/v) and are therefore pro-

Vol.3, No. 2-4 71

portional to the coefficients A2 and A3. An NMR spec-trometer having quadrature detection has tworeceiver channels (Labelled and Xand /channels)and is able to simultaneously observe the signalsproportional to A2 and A,. The Zeeman energy is pro-portional to A, while the quadrupolar energy is pro-portional to A,- The description of the double quan-tum coherence (13) of the system is contained in thecoefficients A and A8. Substitution of Eq. 21 into Eq.14 results in a set of equations for the coefficients

A,(t) = / I { t r [QQ4 ] / t rQ/}A(O(22)

-J [Qj,xr(t)} [Q, e—Kff-r) ei"5cYf-T)e'"Jdr

{Ai(t)-Ar}These equations describe the evolution of the densi-ty matrix during the application of an RF pulse and, ifco, is set equal to zero, the evolution in the intervalsbetween pulses. The relaxation Hamiltonian, Eq. 7,can also be written in terms of the operators Q. Theresulting equation is

+ (Vyy-

2 VxyQ7 ̂ (23)

In the theory to follow, only pulses which satisfythe condition co, » coQ will be considered, and it willbe assumed that the duration of the pulses is shortenough that relaxation during the pulse can beneglected. Under these circumstances, the effect ofan RF pulse can be easily tabulated as has beendone by Vega and Pines (Table V in reference 10).The sudden changes in the coefficients A; that occurwith the application of pulses along the xand /direc-tions are given in Table 1.

Table 1The Effect of a Strong Radio Frequency Pulse

(a) Applied in the X Direction

A2 A2

A, A, cos co, f - A, sin co,fA, A4 cos co,f + A3 sin co,fA A cos 2co, t + (Aa/2) sin 2co, t- (A/2) sin 2co, fA A cos co, t + A7 sin co, tA7 A, cos co, f - A sin co,fA, (A*/4)(3 + cos 2co,0 + (A,/4)(1 - cos 2w,t) -

(A/2)sin2a>,fA, (3As/4)(1 - cos 2co,f) + (A,/4)(1 + 3cos 2cu,0 +

(3A/2)sin2co,f

(b) Applied in the Y Direction

A2 A2 cos co, t + A, sin a>, tA A3A A cos Wit- A2 sin a>,fA A cos co, f - A , sin co, rA A cos 2co, / + (A8/2) sin 2co, t + (A/2) sin 2co, fA, A cos co, f + A5 sin co, fAt (ASIA)(3 + cos 2co,f) - (A,/4)(1 - "cos 2co,f) -

(A/2)sin2co,fA -(3A,/4)(1 - cos 2co,f) + (A/4)(1 + 3cos 2co,f) -

(3A/2)sin2co,f

After a pulse has been applied, the spin systemevolves in the rotating frame under the influence of Vand 3Cr(t). In the particular case where the spinsystem is exactly at resonance and no RF power isbeing applied, the equations of motion of the coeffi-cients A/ derived from Eq. 20 are

dA2l dt= -coQAs - A2I Tla

dA3ldt=uQA6- A3IT2u

dAJdt=(A«- A/*)IT,Z

dAJdt=u)QA2- AsITlb

dAJ dt= -coQA - A / T2b

dA7ldt=-A7IT2D

dA8ldt=-AeIT2D

dA9ldt=-AJTlQ

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

The relaxation times introduced in Eq. 24 to 31 aredefined as follows

MT2a= K[3Jo(0)l4 + 5J,(coo)/2 + J2(2coo)] (32)

1 / T2b = K[3 Jo(0)/4 + J,(coo)/2 + J2(co0)] (33)

1/ r , z = KV,(a>0) + 4J2(2co0)] (34)

1/r,Q=K"[3J,(co0)] (35)

M o ) + 2J2(2co0)] (36)

where

(37)K=2{eqlh(2l--\)]2

These relaxation times correspond to the T22, T66, T33,T77, and T5S introduced by Jacobsen eta/(5).

The spectral density functions in Eqs. 32-36 aredefined as follows

72 Bulletin of Magnetic Resonance

J0(0) = 3/8 JVM) V.,{t+ r)ch

«/.(«») = Yif[VM)+ iVJt)]

[VJt+ T ) - iV,x(t + r)]e"0TdT

/!_>o)1/16 \[Vxx(t) + Vyy(t)] + 2iVxy(t)]

[ !/„( t+T)-Vyy(t+ T)] -2iVxy{ t + T)] e 2 - 0 " "Now that the appropriate relaxation times for a spin 1system have been introduced, the next step is todetermine pulse techniques to measure each one.

It should be noted that the set of Eqs. 24-31 caneasily be generalized to include the extra termsresulting from the application of an RF field and be-ing off resonance, but their solution becomes moredifficult.

B. The Free Induction Decay (FID)

When the spin system is in equilibrium and beforethe application of any pulses, the only coefficient ofthe density matrix is

; 0 = /„ (38)

where pL = 1/ KT. Use has been made of the hightemperature approximation. After the application ofan RF pulse directed along the yaxis in the rotatingframe of width L, the nonzero coefficients are

andA,(0) = /0 cos a

A2(0) = /osin a

(39)

(40)

where a•= &>,£,. Now the evolution of the system isgoverned by Eqs. 24, 26, and 27, and at a time t later

where

and

A,(0 = /o[1 - (1 -cos a) e-'Tlz]

A2(t)= /oSina e"T2{cospt+ 1/

A5(t) = los\na e"T2{wQlps\n t}

MT2 = V2 [1 / T2a + 1 / T2b

1/y = V2 [MT2b + MT2a

Usually coQ2»1/y2 and

A2 = l0 sin a er"T2 cos a>Qt

As =• /0 sin a e"T2 sin ojQt.

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

In the case where a = n/2, the nonzero componentof the density matrix oscillates between A2 and A5 inthe absence of an RF field.

The observed signal following a (n/2),, pulse will beproportional to A2 given by Eq. 47 and an analysis ofthe decay of this signal can be used to obtain T2. TheFourier transform of the FID is the frequency spec-trum. Figure 1a shows a typical FID for an orientedsample and the corresponding quadrupolar splitspectrum.

In powder sample, where there is a range of valuesfor a>Q, the decay of the FID is a result of the dephas-ing of the various signals from the individualcrystallites in the sample as well as spin-spin relaxa-tion. In a powder, the observed FID is proportional to

j ' e-"2 cos {AQ{V2(3M2 - 1)}774*

where \x = cos 6 and 0 is the angle between the axisof symmetry of the electric"field gradient and thedirection of the applied magnetic field. Figure 1bshows a typical powder pattern FID for an unorientedsample. Notice that the first oscillation in the FID hasa much larger amplitude than the succeeding ones.The Fourier transform has two well defined peaksseparated by AQ.

C. The Quadrupolar Echo

The nuclear signal from deuterons is generallyquite weak. Experiments usually require the use of ahigh-Q coil to enhance the sensitivity, with the resultthat there is a long receiver "dead time" followingan Rf pulse. This dead time may be as long as 30 ̂ s,making it impossible to get an undistorted frequencyspectrum directly from the FID. This is especiallytrue for a powder sample where the initial part of theFID contains information about the wings of thespectrum. Because the quadrupolar splittings areassociated with the time-averaged or static-quadrupolar Hamiltonian, and because / = 1, it ispossible to circumvent the receiver dead time pro-blem by using a two-pulse sequence which results ina quadrupolar echo (9). The pulse sequence consistsof a (n/2),, pulse followed at a time T later by a (n/2),pulse. At another time T later, a refocussing of themagnetization occurs. The amplitude of the echodecreases with increasing T, however, because of ir-reversible losses of phase coherence due to staticmagnetic dipole interactions with other nuclei andfluctuating quadrupolar and dipolar interactions. In

Vol.3, No. 2-4 73

oo

N

cF5o

0)<Q3CD

o'DOCDwOu0)oCD

O L

o

ro

Q)

Figure 1. (a) The free induction decay and frequency spectrum of the deuterons in an ordered sample consisting of 30%potassium laurate-ad2, 60% H20, 6% decanol, and 4% KCI at 16°C. (b) The free induction decay and frequency spectrum ofthe deuterons in a mixture of 70% rubidium stearate-ad2 and 30% H20 at 92°C. The free induction signals were, in fact,recorded starting at the peak of the quadrupoiar echo to avoid the receiver dead time.

the theoretical outline given here, the magneticdipolar interactions have been neglected and onlyrelaxation due to the time-dependent quadrupolar in-teraction is considered. The nonzero coefficientsafter the first pulse are given by Eq. 47 and 48. Im-mediately after the (71/2),, the nonzero coefficientsare

A2(T) = cos

A5(T) = k e T / " sin

(49)

(50)

The density matrix develops according to Eqs. 24and 27 and the observed signal at a time t after thesecond pulse is proportional to

A2(t) = k e '• COS coQ(t— T) (51)

The echo which occurs at t= T is independent of coQ

and is therefore very valuable in the study of powdersamples. A typical quadrupolar echo signal is shownin Figure 2. The signal recorded from the center ofthe echo onward is a reproduction of the FID reduc-ed in amplitude by the factor e2 ' "2 . The Fouriertransform of this signal gives an undistortedquadrupolar split powder pattern spectrum.

The FID shown in Figure 1b was in fact obtainedusing the echo technique. Generally, the first largeoscillation in the FID is lost in the receiver dead time.

Figure 2 shows a sequence of echoes for a powdersample. Measurement of the amplitude of the echoas a function of T can be used to measure T2. If T2

depends upon the orientation of the electric fieldgradient with respect to the magnetic field, anestimate of the functional dependence can be ob-tained from an analysis of the FT spectra for a se-quence of T values. Figure 3 shows an examplewhere T2 is dependent on orientation. The shape ofthe powder spectrum changes as the spacing bet-ween the two pulses increases. The value of T2 takenfrom the decay in amplitude of the peaks in the spec-tra is 20% less than the value taken from near thecenter of the spectra.

D. Measurement of TIZ

When any RF pulse is applied to the spin system, anonequilibrium value of the Zeeman energy results,which means that A, has a value different from l0.Then, to measure TIZ, A4(t) is monitored as a func-tion of time. The most commonly used sequence ofpulses is the (n), - T - (nl2)y sequence. After the firstpulse

A = -/o (52)

•\A300/<s

\M

'\AAA / \ / \ / \ / \ /500 yuS

0 1ms

Figure 2. The quadrupolar echo in a powder sample as a functionof spacing between the two pulses. The lyotropic liquid crystalsample was a mixture of 70% rubidium stearate-ad, and 30% H2Oat92°C.

and at a time T later

(53)

The amplitude of the FID following the n/2 pulse ap-plied at time T is proportional to A,(T). Frommeasurements of FID amplitude as a function of T, 7",can be obtained. In a powder sample, the echo se-quence is used in place of the nil pulse. By Fourier

Vol.3, No. 2-4 75

CD

oo

CDc

5o

03CO3CDo'3JCDCO

o30)3OCD

O LO

Figure 3. The Fourier transforms of a sequence of quadrupolar echoes recorded at different spacings (T) between the twopulses. The spectra of the a deuterons in a stearate chain in a mixture of 70% rubidium stearate and 30% H2O were recordedin the gel phase at 16°C. Notice the change in the shape of the powder pattern with increasing T, indicating that T2 dependson the angle betwen the electric field gradient and the magnetic field.

transforming the echo signal for each T value, theangulardependence of 7"1Z can be obtained.

E. Creation of Quadrupolar Order

In order to measure 7"1Q, it is necessary to create anonzero quadrupole energy, that is, a nonzero valueof the coefficient A,. The sequence (nl2)y - T, - (n/4),results in the following nonzero coefficients im-mediately after the second pulse

A2(TI) = /o e T l / r 2 COS a>0T,

/ " sin

(54)

(55)

(56)

With an oriented sample, it is possible to apply thesecond pulse such that cos coeT, = 0 and T, « T2. Us-ing this sequence, it is possible to transform theoriginal Zeeman order into a mixture of quadrupolarorder and double-quantum coherence as signifiedby nonzero values of A, and As. In a powder, it is notpossible to satisfy the condition weT = nil for all do-mains in the sample. If this sequence is used, therewill always be a variation in the quadrupolar orderacross the sample.

F. The Observation of Quadrupolar Orderand Measurement of T1Q

After the creation of quadrupolar order, thenonzero coefficients of the density matrix evolve ac-cording to Eqs. 24,26,27,30, and 31. The coefficient A,will decay with characteristic time TlQ. At a time Tafter the creation of the quadrupolar order, thenonzero coefficients are

A, (T) = / „cos OJQTI e ' 1 / r 2 c o s WOT

A, (T) = /O(1 - e T l / r i z )

AS(T) = kcosCOQT, e T l / " s in WQT

A>(T) = ( / O / 2 ) s in COQT, e-rUT2D

(57)

(58)

(59)

(60)

(61)

where it has been assumed that-r,/ 7"2«1.

Since the coefficient A, is not directly observable,a second (n/4), pulse must be applied in order to con-vert A, into A5. At a time f after the (n/4), pulse

A2(t)= cos , cos

sin

e"r2cos

, e"T1 sin wQt(62)

- e-T/riz)er /r2cosco0f (63)

eT/r 'z) (64)

As{t)= cos cos wQT e" r 2sin u>Qt-

(/0/4)(eT/r2D + 3eT/riQ) sin wcTl e ' / T 2cos(65)

In an oriented sample where there is a single valueof coQ, it is possible to make weT, = n/2 so that

-T/7"2O _l Q O-T/riO n i r t t / dd\

~r 0 6 v SID COQI \0D;

-25 0Khz

25

Figure 4. (a,b) The nuclear signals observed in the two detectorchannels following the second (n/4), pulse in the Jeener -Broekaert sequence, (nl2), - T, - (n/4), - f - (n/4),. The signals are90° out of phase. (c,d) The Fourier transforms of each signal takenseparately. The Zeeman order in the spin system is proportion tothe amplitudes of the signals in (a) and (c) while the quadrupolarorder is proportional to the signals in (b) and (d).

Vol. 3, No. 2-4 77

r ms

Figure 5. The decay of the signal as a function of T for the (n/2)v - T,- (n/4), - T - (n/4), sequence, T, was held constant at 22 ^s. Thedecay follows the predicted behaviour. It is proportional to (e-"r2D

+ 3eT/ri°). The ordered sample was a mixture of 30% potassiumlaurate-ad2, 60% H2O, 6% decanol, and 4% KCI at 6°C.

A,(f ) = -(/0/v^)(1 - er/Tlz) e' /r2coswQf (67)Notice that the two signals which oscillate with fre-quency coQ are 90° out of phase. The signal in the Xchannel which is proportional to A2( t) is zero at /= 0,in contrast to the FID following the first (n/2), pulsewhich has a maximum at f = 0. Figure 4 shows thesignals in the two detector channels for the orientedsample as well as the Fourier transforms of thesesignals. Zeeman order is characterized by the usualdoublet spectrum, while quadrupolar order results inan absorption signal for one peak and an emissionsignal for the other. By measuring the amplitudes ofthese two signals at a fixed time tand as a function ofT, the relaxation times T2D, TlQ, and TIZ can be deter-mined.

The interesting prediction of Eq. 66 is that onequarter of the signal should decay with a time cons-

tant T2D and three quarters with a time constant TIQ.In Figure 5, the amplitude of the signal in the Xchan-nel is shown as a function of the spacing (T) betweenthe two (n/4), pulses. The decay is definitely the sumof two exponentials. The amplitudes of the two com-ponents are approximately in the ratio of 1 to 3 as ex-pected. The measured values of T2D and T,Q at 6°C forthe ordered sample were 440 ̂ s and 43 ms, respec-tively.

In a multi-domain sample, Eqs. 62-65 must be con-sidered. The first term in the expression of A2 couldcomplicate the measurement of T2D and TlQ. Thespin-spin relaxation times T2 and T2D are, however,often much shorter then TlQ, so that the first term inthe expression for A2 can be neglected. Observa-tions of A2 and A3 in this situation can be used to ob-tain f,oand TIZ.

78 Bulletin of Magnetic Resonance

Because of the dead time of the receiver,measurement of the entire FID is difficult, precludingany chance of getting an undistorted signal followingthe s e c o n d (nIA), p u l s e . C o n s e q u e n t l ymeasurements of TiZ and T1Q as a function of anglefrom the powder pattern are difficult. The problem ofreceiver dead time can be overcome by applyinganother (nl2)x pulse to create an echo. If the (n/2),pulse is applied at time t,, then the essential coeffi-cients at a time f later are

A2(t) = / o e T / "

coswQ(t- t,) +(/0/4)(e-' / /2D + 3e " " Q )

sin a)0T,e (" *lUT2 sin a>Q(t- A)

(68)

and3(t) = (lol\f2)(1 - eT /" z) e"ncos«oQt (69)

The second term in the expression for A2 (t) is ofprincipal importance. The signal arising from thisterm for times t> tt is identical to the signal followingthe second (n/4), pulse, but now the effective start ofthe signal is at / = f, and is not lost in the receiverdead time. Figure 6 shows that changing tv effective-ly moves the beginning of the signal progressivelyfurther out from the (nl4)x pulse. It should be notedthat the beginning of the signal (t = f,) is at a zerocrossing because of the form of Eq. 68. As long as T2

and 7"2D are relatively short compared to TlQ, theterms containing e T / " and e-/T2D will not interferewith the measurement of T,Q which is made by vary-ing T and measuring the amplitude of the signal inthe ^channel of the receiver.

In situations where it is expected that TlQ dependson the angle between the electric field gradient andthe static magnetic field, it is useful to Fouriertransform the signal and measure TlQ for severalvalues of wQ. It is possible to start digitizing thesignal at t= tu then Fourier transform to obtain a fre-quency spectrum. The shape of the spectrum issimilar to the derivative of the usual quadrupolarpowder pattern. It was pointed out previously that ina powder sample it is not possible to transfer theZeeman order entirely into quadrupolar order, hencethe factor sin COQT, in Eq. 68. Because of thedependence on sin COQT,, the Fourier transformedsignal will have a complicated shape reflecting thefact that the conversion from Zeeman to quadrupolarorder varies across the wide frequency spectrum.The conversion is maximized in the part of the linewhere WQT, = nil. Figure 7 shows the change in the

50>s

500>s

Figure 6. The effect of changing the spacing t, in the sequence(n/2), - T, - (n/4), - T - (n/4), -1, - (n/2),. The effective beginning of thesignal (indicated by the dot) is moved progressively further awayfrom the final (n/4), pulse as t, is increased. The powder samplecontained 70% rubidium stearate-ad2 and 30% H2O held at 92°C.

shape of the nuclear signal with T,. The Fouriertransforms of these signals starting at t = tt areshown in Figure 8.

The angular dependence of T1Q can be obtained bycollecting spectra for a variety of values of T at a fixedT,. The decay of the amplitude at a number of dif-ferent frequencies can be analysed to obtain thedependence of TlQ on «Q, which in turn depends onthe angle 0. In Figure 9, the Fourier transformedsignals for a number of T values are shown. An

Vol.3, No. 2-4 79

Figure 7. The effect of changing T, in the pulse sequence (n/2), - T,-(n/4),-T-(nlA)x- f, - (n/2),. As T, changes, the condition COOT, = (2N+ 1)(n/2) is satisfied at different parts of the powder pattern. Themost efficient conversion of Zeeman into quadrupolar order thenshifts to a different part of the line, causing the change in theshape of the echo. Echo signals are shown for several values of T,.

analysis shows that T1Q varies about 20% across thefrequency spectrum.

While the signal following this pulse sequence canin principle be used to obtain both T1Q and 7"2D, itwould be useful to have a method of measuring T2D

separately since only one-fourth of the signal decayswith a time constant T2D. A modification of theprevious sequence removes the dependence of thefinal signal amplitude on T,Q.

G. Creation of Double-Quantum Coherenceand Measurement of T2D

One pulse sequence which can be used tomeasure TiD is (nl2)y - T, - (nlA)x(nl2)y - T - (TT/4),. The

nonzero coefficients of the density matrix followingthe first (nIA), pulse are given by Eqs. 54-56.

The applicaton of a (71/2), pulse immediately afterthe {n/4)x pulse eliminates the quadrupolar order, orA, component of the density matrix. After the (n/2),pulse, the spin system evolves in the absence of anyRF field according to Eqs. 26 and 30 for a time T. Atthat point, the nonzero coefficients are

A ( T ) = /0{e-"T1 cos WQT, - 1 )e T / r i z + 1} (70)and

s in COQT, e-T/ (71)

It is the decay of the As coefficient with T which is tobe determined but, in order to monitor the decay,this coefficient must be rotated into one of the coeffi-cients that can be directly observed. Such rotationcan be brought about applying another (nlA)x pulse.The observed signals which appear in the two chan-nels of the receiver are proportional to

andsin COQT, e-T/r2°sin w o f e I / r 2 (72)

(73)cos COCT, - 1) + 1} cos w01e

In an oriented sample, it is possible to make COOT, =rr/2.Then

andA,(f ) =

e-1"2 eT /""sin coQte-"2 (74)

e-""*)cosa>sfe-'"2. (75)

Notice that both observed signals oscillate withangular frequency wQ and are 90° out of phase.

By measuring the amplitudes of these signals atan appropriate value of fas a function of T, it is possi-ble to obtain values for T2D and TIZ.

In a powder sample, a wide range of a>Q values ispresent, most of the signal will be obscured by thereceiver dead time because of the dephasing due todifferent u>Q values. The application of another (n/2),pulse a time U after the second (n/4), pulse producesan echo in the A"channel at a time t, later. The echosignal is proportional to

A2(t) = 2° sin

sm uQ(t-U) (76)

It should be noted that this particular sequence ofpulses results in a single term proportional to e'T1D

even in the case of a powder sample. A value of T2D

can be obtained from measurements of the echo

80 Bulletin of Magnetic Resonance

CO

i

OO

3N O

Oo

Figure 8. The Fourier transforms of the echo signals in Figure 7. The echo signal to be transformed was started at the dot in-dicated in the traces in Figure 7. As T, increases, the effect of the factor sin CO,T, is very evident, The powder sample contain-ed 70% rubidium stearate-ad2 and 30% H2Oheldat92°C.

00

Figure 9. The decay of the quadrupolar order in a powder sampleconsisting of 70% rubidium stearate-adj and 30% H,O held at 92°C.The value of Tw varies about 20% across the spectrum.

amplitude as a function of T. If T2!) is dependent onthe angle between the electric field gradient and themagnetic field, then Fourier transforming the signal,starting at the center of the echo, and measuring T2Dfor various values of a>0 will give an indication of theangular dependence. The shape of the echo and theobserved powder spectra obtained from Fouriertransforming the echo will be similar to thoseobserved in the measurement of TiQ and shown inFigures 7 and 8.

III. CONCLUDING REMARKS

The possible relaxation times for a spin 1 systemwith a Zeeman-plus-quadrupolar interaction havebeen discussed along with the various pulse se-quences needed to measure these times. Thediscussion started with a particularly simple Hamilto-nian and the spin system was allowed to evolve in

the absence of an RF field and in the on-resonancecondition.

As a first step toward extending these results, itwould be interesting to study the equations for thecoefficients A,{t) in the presence of an RF field.Measurements made with the field on would corres-pond to the familiar 7~lp measurements presently us-ed to advantage in spin Vz systems. It is a relativelyeasy task to set up the appropriate equations for theAt(t) with an RF field on (5), but their solution will bemore difficult.

The next step in extending this work would be toconsider other terms in the initial Hamiltonian give inEq. 1. In particular, the dipole-dipole coupling bet-ween deuterons and currounding protons will be,im-portant in most practical applications of this work.There are two aspects to consider. The secular, ortime-independent, part of this dipolar interaction willcause a dephasing of the spins as they precess, inmuch the same way as happens when the staticmagnetic field is inhomogeneous. It is probably thiseffect which determines the values of T2 and Tinmeasured for the two samples used as demonstra-tions of the pulse techniques in the previous sec-tion. The time-dependent part of the dipolar interac-tion will also be a relaxation mechanism and couldcontribute to the observed spin-lattice and spin-spinrelaxation times in certain circumstances.

REFERENCES

'J. Seelig. Quart. Rev. Biophys. 10. 353(1977).2H.H. Mantsch. H.Saito, and I.C.P. Smith. Prog. NMRSpectr.n,

211(1977).'J.H. Davis. K.R. Jeffrey, and M. Bloom. J. Magn. Resort. 29.191

(1978).4M.F. Brown, J. Seelig, and U. Haberlen, J. Chem. Phys. 70, 5045

(1979).M.P. Jacobsen. H.H. Bildsoe, and K. Schaumberg. J. 'Magn.

Reson. 23.153(1976)."J.P. Jacobsen and K. Schaumberg, J. Magn. Reson. 24. 173

(1976).'R.R. Void and R.L. Void, J. Chem. Phys. 66, 4018 (1977)."A.G. Redfield, Adv. Magn. Reson. 1.1 (1965)."J.H. Davis, K.R. Jeffrey. M. Bloom. M.I. Valic. and T.P. Higgs.

Chem. Phys. Lett. 42, 390 (1976).'°S.B. Ahmad and K.J. Packer, Mol. Phys. 37, 47 (1979)."S.B. Ahmad and K.J. Packer, Mol. Phys. 37. 59 (1979).c iJ. Jeenerand P. Broekaert, Phys. Rev. 157, 232(1967)."S. Vega and A. Pines-1. J. Chem. Phys. 66 5624 (1977)IJR.C. Long. J. Magn. Reson. 12 216 (1973).ISR.A. Hoffmann, Adv. Magn. Reson. 4, 84 (1970)."A. Wokaunand R.R. Ernst, J. Chem. Phys. 67. 1752(1977).

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82 Bulletin of Magnetic Resonance