PHYSICAL REVIEW B VOLUME 8, NUMBER 9 1 NOVEMBER 1973

Nuclear Spin-Lattice Relaxation in CaF, via Paramagnetic Centers for Short CorrelationTime when Spin Diffusion is Inhibited

N. A. Lin and S. R. HartmannColumbia Radiation Laboratory, Department of Physics, Columbia University, New York, ¹wYork 10027

(Received 26 June 1973)

Nuclear spin-lattice relaxation, in the presence of an intense rf field H, at frequency w shifted fromexact resonance coo = y Ho so that first-order spin diffusion is inhibited, has been studied in theshort-correlation-time limit coor, «1. The experiments satisfy the magic-angle condition [cos'(I5 = 1/3,where tan P = H, /(Ho —co/P)], which eliminates the first-order spin-flip term I, I, from theeffective Hamiltonian. When 7; is not so short that higher-order spin diffusion is unimportant, thenuclear spin-lattice relaxation has the long-term behavior S(t) ~ B exp[ —(t/r)'~ ]. When r, is shortenough that the higher-order spin diffusion cannot be neglected, the magnetization relaxes toward anequilibrium value according to S(t) = exp( —t/T»). Experiments show T, ~ A H,'", where H, isthe effective field. Theoretical consideration shows that, to the lowest order in H„ the higher-orderspin-diffusion constant has the form D

p= K/H „where K is a constant. With this form of diffusion

constant, the spin-lattice relaxation time will have the form T lp (C p) (D p) = A H,'" in thediffusion limited region, as is obtained in the experiments.

I. INTRODUCTION

The theory and experiment described here con-cern the spin-lattice relaxation (due to paramag-netic impurities) of a nuclear spin system in thepresence of a strong, oscillating, nearly resonantmagnetic field. The spin system investigated isa rigid array of identical spin-& nuclei, coupledonly by their magnetic dipolar fields.

When transformed to a frame rotating at thefrequency e of the rf field, the effective field ofthe spin system is static and is given by

Hg = H,x+ (Ho —(d/y) z yi

where Ho is the static dc field, H, is the amplitudeof the rotating rf field, and y is the nuclear gyro-magnetic ratio. ' The effective Hamiltonian in thisrotating frame is given by

2

&eP =@ —H IZ+N~ 2

with

~.=yH, = [(~0 —&u)'+(ufj'",

&o(p) = z(3cos p —1),&,q(p) = —z sing cosy,X„(y)= 4 sinzrp,

%= ~ B ~(I zI~z - -'&i '&g)i&j

+ = 2 B;i(I(+I)z+IizI,,),i&j

+ = Z B;J(I;,I;,),3C, = (K,)

X,=(K)~,

where

3y S (1 —3 cosz8;, )fj rs

t jIn this rotating frame, the secular part of the

dipolar interaction (the part which conserves Zee-man energy) is reduced by a factor of —,(3cos y —1)from its value in the laboratory frame. By suit-ably choosing the amplitude II, and frequency cu

of the oscillating field, we can satisfy the magic-angle condition cos p = ~, where tang =H, /(Ho —~/y), and the first-order spin diffusion maybe reduced to zero. The nuclear spin system canthen be considered to interact only with the para-magnetic impurities if the relaxation effect of theimpurities is strong enough that the residual spin-spin interaction can be neglected completely. Sucha diffusionless relaxation has been studied in theregion where ~os-, »1. The first part of the pres-ent investigation is a study of the relaxation for+pT ~~ 1 while the spin diffusion is still negligible.

The magic-angle condition does not eliminatespin diffusion altogether, as multiple-spin-flipprocesses can take place by way of the I;,I&.,I;gj~, etc. , terms in the effective HamiltonianX„. When the noise power generated by the para-magnetic impurities is not strong enough for us todisregard these multiple spin flips, the residualspin diffusion may play an important role in therelaxation of the distant nuclei as a result of theircoupling (through spin diffusion) with those nucleiwhich are close to impurities sites. The secondpart of the present investigation is a study of theeffect of the higher-order spin diffusion on thespin-lattice relaxation .

4079

4080 N. A. LIN AND S. R. HAR TMANN

II. SPIN-LATTICE RELAXATION WITHOUT HIGHER-ORDERSPIN DIFFUSION

8 * = —Q Wz»M«„,f

respectively, where

w„„=, -s(s+1)(y,ya)'1 1

fk

(3)

(3cosa»sin8») 'z z1+(up ~c

Wz~« ——p

—S(S + 1}(y&yK)1 1 2

fk 3

x 1 —3cos &fk 2 21 + Q) e 7'c

(4)

For negligible spin diffusion and Hp» H~

»((&H)z„«)'tz, the magnetization relaxation alongthe dc field and the rotating rf field can be writtenas

8M~ = —Q W~»(M«, ™«p)

I1 —3cosz& I

&sWP

Equation (7} shows that the magnetization of anindividual spin polarized along the effective fieldwill relax as exp(- g&t/TU«) in the absence of spindiffusion. This will differ from the relaxation be-havior of the total magnetization, since the re-laxation rate T»k depends strongly on relativeposition. In fact, nuclei within a certain regionsurrounding each impurity have resonant frequen-cies which are shifted outside the main resonanceline because of the effective static magnetic fieldfrom the impurity. Those nuclei, therefore, donot contribute to the observable magnetization.In calculating the spin-lattice relaxation of thenuclear magnetization, these regions have to beexcluded. Such a region can be defined in termsof a critical distance p from an impurity site atwhich the effective static field due to the impurityion equals the local nuclear dipolar field Hd of theF' sites. Taking into account the angular varia-tion of the effective dc field, we have

+ z (3 cos8„sin8, «)z1+~07c (6)

kZ

Hp —Pl/y

From Eqs. (2)-(6), we obtain

eM„, 1«z(M«z —M«„),

&~fk

with

1T

= Wefk+ Wpfk1fk

C,(1 —3 cos'fI»)' C, cos'8» sin'8»e + 6

+fk fk

where

(6)

C, =p S(S+ 1)(y«y8')

+ &e7'c

C(l = 2 S(s+1)(y«y~)' 1+~2~2

(6)

M«ee = « &3 M«p Q Wp J«g(W((» + Wp»)J

and we have used the magic-angle condition cos pSin +=3 ~

and Mkp is the magnetization at thermal equilibriumin the absence of the rf field H&. If we let themagnetization along the effective field H, be M~,then we have

—zv 3 duE W(l, «

&exp —Z(W„„+Wp, «)(u+ t) (12)

where V, is the critical volume, N~ is the impurityconcentration, and g implies that the summationis carried out only for nuclei outside the criticalradius. Using the technique of FGrster to evalu-ate Eq. (12), we obtain

1/3= p. Il -3co"6I"',

d

(11)where (t(&) is the effective magnetic moment givenby Rorschach. ' Dividing by the total magnetizationat t = 0, and considering the overlapping probabil-ity of the critical volumes, we obtain the normal-ized relaxation equation of the magnetization inthe form

St = M(t) —M(~ )

yHp

=+'(M, (0) —M, ]eee -Z(W, e, ~ (ee, le)k f

x Mkc{0

=e» exp — {Wefk+ Wpfk)t

S(t) = exP( —(2vt'tz/3)[4'~(t) +4(z(t)]) —(7( ~z/2v 3)CP«' ggz'

(1 —x )x dx3p)'+C (1-x')x']"' ' Qv3V,

"' " ' (1-3x'}'

NUCLEAR SPIN- LATTICE RELAXATION IN. . . 4081

x exp[ —2v(u+ f)"'[e,(u+ t) + +z(u+ t)]/3}, (13)

with~+1 18' 1-x' x'e, (t) =X,Wv ' [C,(1 —3x')'+C, (l-x')x']"'p C, +C,"1 C

where p1 is the error function.ln the limit &nor, «1, Eq. (13) has a long-term

behavior

S(t) =e "'~' (1 v3 p/a),where the constants are defined by

a= f, [2(1-x )x +2(1 —3x ) /9]'t dx=1. 30,

(1 —x)x dx[2(1 —x')H+ 2(1 —32)z/9]'~z

S(t) =expj- (t/r ) [4 (t)+4 (t)]}, (14)

1/r~ = ', vr N~(y~—yR)[S(S+ 1)T,] a

The intercept of the long-term behavior at t =0 is

S(0) = (1 —W3P/a) = 0. 483 .When the correlation time of the electron spin

is long enough that +Or, »1, then because coo»co„Eqs. (8) and (9) show that the low-frequency noiseis more effective in relaxing the spin magnetiza-tion than is the high-frequency noise. In thiscase, the equilibrium magnetization along the ef-fective field may be taken as zero, and the secondterm in Eq. (13), which results from high-fre-quency noise, may be omitted. Using C, »CO, weobtain the relaxation equation for the case (d07» 1as follows:

III. SPIN-LATTICE RELAXATION WITHHIGHER-ORDER SPIN DIFFUSION

The spin-diffusion coefficient due to the seculardipolar interaction was studied by Redfield in thehigh-field case, where he used the truncated Ham-iltonian of Van Meek. For the lower-field case,such truncation is not suitable, and the effect ofnonsecular dipolar interaction may play an im-portant role, especially when the secular termsare somehow quenched.

The manner by which the nonsecular terms inEq. (1) cause spin diffusion is made evident if weconsider the transformation" p'=e ' p', whichleads to the equation of motion

dp z

a [p

with

x'=e "x,pe" =x, +x, /(u, ,

where

[x„x,]=0,

X, = - tt-~.Iz,

Z —x„(y)x„,M

where we have X,=e(zz, [X „X,]+Xz[X „Xz]/2} (18)

18' (C.t)'t z

&3 VC

@() (,), 2562 (c. )

1 161(p

=~3

7t' N~C

Equation (14) should replace the result of Ref. 3,where an incorrect form of the critical radius wasused. ' When (C,t) 'tz/V, »1, we may replace @,(t)with unity and +z(t) with zero. We then have thelong-term behavior for diffusionless relaxation:

S(t) e-«/~, )' '

In this limit there is perfect agreement with Ref.8

for sufficiently high effective field. " With theabove transformation we obtain I,~ =e ' I&~e'=I», since the effective field is large, and there-fore we find (I,z) = Tr[I&zp'}, where p (t) =e '

& p (0)e'" " . Since X' contains the mutual spin-flip term in the form I;,I, I„z /up, in first order,we expect that the resulting diffusion constant D„due to the nonsecular dipolar interaction, will beinversely proportional to cu, .

A detailed derivation of D, using the techniqueof Redfield is given in the Appendix. Aside fromthe dependence on co„ the expression for Dp wastoo complicated for us to evaluate.

In order to evaluate D, experimentally we mustrelate it to T».

Combining the effects of direct relaxation and

spin diffusion, we find that the magnetization pc-larized along the magic angle obeys the equation

4082 N. A. LIN AND S. R. HARTMANN

=D, V'M(r f) — ' [M(«)-M ]

(17)where we have used the single-paramagnetic-cen-ter model and have taken the angular average overthe quantities which are angularly dependent. Ex-plicitly, we have

C, =C, ((1—3cos'8)~)+C, (cos 8sin 6)= 48(S+ 1)(ypyh )ar, /9,

&3 Co(cos26 sin 8)MO2 C, ((1 —3cos 8) )+Co(cos 8 sin 8}

= 9MO/10& 3,where ~pT «1 is assumed.

The same form as Eq. (17) also exists for re-laxation along the static field and the rotating rffield. A number of authors" used suitable bound-ary conditions to solve Eq. (17) and obtained therate of increase of magnetization per unit volumeper unit time in the form

I 0-

0.483 7

O. I

O.OI

SAMPLE: COF2

Np(~m j 4 5 X ~0 fons/cm3+ . 20 3

(18)

with

1 r(-,')f„,(8,)="'"""r( ')I"'(3)- (19)

where P, =(C, /&, )' and 8, =(P, /h, ) /2. The quan-tity b, is called the bqrrier radius for diffusion.Although the spin-lattice relaxation time in theform of Ea. (19) exists for relaxation along thestatic field, rotating rf field, and magic-angle ef-fective field, the diffusion constant and the relaxa-tion parameter are different for each case. %eshall use the quantities without the indexes r and

p as those related to the relaxation along the staticfield; we shall use index x to denote relaxationalong the rotating rf field; and we shall use the in-dex p to denote relaxation along the magic-angleeffective field.

IV. EXPERIMENTAL RESULTS AND DISCUSSION

The measurements were made using CaF& dopedwith rare-earth ions obtained from Optavac, Inc.The static applied field was 5900 G, and the CaF2crystals were immersed in liquid nitrogen. Forthe measurement of spin-lattice relaxation in themagic-angle rotating frame, the technique of Tseand Hartmann was used, while for the measure-ment of spin-lattice relaxation time in the labora-tory frame, T, , and in the rotating frame, T,„,standard methods' were used.

In order to reduce the effect of higher-orderspin diffusion and to satisfy the condition (dpT «1,CaF2 doped with a high concentration of TmF3was used. The theoretical result (solid line) from

O.OOII I

2 3 4t "2 (msec)~/2

FIG. 1. ln of the magnetization which is polarizedalong g~ (at the magic angle) plotted as a function of t /

for cuoT~&& 1.

Fq. (13) and the experimental data of this diffusion-less relaxation are shown in Fig. 1. The long-term behavj. pr pattern e '/' and the flat initialdecay are the same as those for (dp, »1. Themain difference between these two cases is the in-tercePt at 1=0. In the case ~pT, »1, the extraP-olation from the long-term slope has an interceptof 1.0 at t=0; in the case (dpT «1, this interceptis 0.483. This is because in the case of (dpT «1,the noise oscillating at frequency ~p =yap in thedirection perpendicular to the static field Hp isimportant in the rel.axation and establishes a,n:

equilibrium magnetization proportional to Hp inits direction. In the presence of a strong rf field,the magnetization precesses about H, in the rotat-ing frame and results in a large projection in thedirection of the effective field.

The theoretical curve shown in Fig. 1 was ob-tained by setting T,' =0.99 msec~ and V, M~=0.21;the latter was obtained from the expression

& = 0.81, which is equal tp the ratjp pf the

NUC LEAR SPIN- LATTICE RE LAXATION IN. . . 4083

300—

200-

E—~ l00-

50-

IO 20He (gauss)

~yQ

yo cO

qo'

I I I

30 40 50

D, =K/H, and I.,«(5,)/I, s«(5, ) are field dependent.An H, ~ dependence of T» implies that I.s~4(5, }/I,,«(5,) is field independent and can be equal onlyto unity, according to Fig. 1 of Rorschach. ' Thuswe conclude that the spin-lattice relaxation is in

the range of the diffusion-limited region.We do not evaluate the constant K in Eq. (A4)

explicitly. Instead we obtain the ratio K/D (D isthe ordinary first-order spin-diffusion constant inhigh field and is readily calculated' and mea-sured' ) from measurements of the ratios T„/T,and T»/T, . Based on Eq. (19), the ratio of T,„/T,may be written

FIG. 2. Field dependence of T~~. Impurity concentra-tion are in mole %.

Is t 4(5) I-s &4(5r}

-si4( s/4( (20)

normal free-induction decay signal amplitudes ob-tained with the doped and undoped CaF~ crystals.Both H, = 17 and 25 G yield the same value of 7

&

indicating that higher-order spin diffusion is notimportant in relaxation. The theoretical value of

may be obtained by the following data: Tg Cp K H, I 3]4 i5(21)

Lowe and Tse obtained 5„=1.535 and (C/C„)'~4x(D/D„}s~'=1.62. Using the fact that T« is in thediffusion-limited region, i. e. , I s ~ ~(5,) = Is «(5,),we write

r, =3.2x10 "sec at 77 'K (Ref. 13),&s =4. 5x10s' ions/cms,

ysK S(S+1)[Tm ']=4.6x10 (erg/G)

which results in v', =0.48 msec', as comparedwith the experimental value of 0. 99 msec'

To study the effect of higher-order spin diffusion,we used CaF~ doped with CeF3. Two crystals wereused: one doped with 1.6 mole/o CeFs, and theother doped with 1.0 mole% CeF3. The values ofT» were measured for all three crystal orienta-tions such that [(&H)„,/H, ]«1 is approximatelyvalid.

In Fig. 2 the measured values of lnT„are plot-ted against lnH, for three different directions; allindicate that the slope is —, (a least-squares fitgives the average slope as 0.770+0.025). In theexpression of spin-lattice relaxation T» in Eq.(19), C, being field independent for ursr, «1, only

The measured values of T,„/T, allow a deter-mination of the ratios Is~4(5)/I s~4(5). Putting themeasured values of T»/H, ' ', T, , Is&4(5)/I s~4(5},and the angular average of C and C, into Eq. (21),we obtain the ratios K/D as shown in Table I.

The measured time constant ~&' for diffusion-

less relaxation is 0. 99 msec', which is largerthan the theoretical value by a factor of 2. Usingthermodynamics, Osiko' showed that the effectiveimpurity concentration is reduced as the nominalconcentration is increased. Using the smallereffective concentration, one can improve theagreement between the theoretical and experi-mental r,'~s. Our measurement of K/D dependson the ratios T,„/T, and T»/T, and is independentof the impurity concentration as well as the cor-relation time The re. sulting K/D should be fairlyaccurate. Our results show that K/D= s G. ForH, = 5 G, this implies that D,/D = ~M, and that D,decreases as H, increases.

TABLE I. Ratio of K/D for three crystal orientations.

Impurityconcentration

(mole %)

Crystalorientation Tf (msec) T$g/Tf

T„/a,"4(m sec/G )

K/D(6)

CeF3CeF3CeF3

1.6fp

1.0/p

Hp[) f111]Hp [[ [110)g, ii [100)

17.223. 2

12.7

1.291.241.38

200/9. 95200/9. 4200/14. 5

0.500.610.64

4084 N. A. LIN AND S. R. HARTMANN

APPENDIX

To obtain an explicit form for the diffusion con-stant due to the nonsecular dipolar interaction inthe rotating frame, we use the method of Redfield.At t & 0, a space-time varying field is added to aspin system in addition to the usual static andstrong rf field, so that the Hamiltonian in the ro-tating frame has the form

XNq 3Cep+ ek(dqZ slIlkx)I)2

where we have assumed that the magic-angle con-dition 3cos p —1=0 has been satisfied. The quan-tity & is set at zero for t&0. With this kind ofHamiltonian, the nonuniform magnetization, whichwill decay for t &0, may be written in the form

(I;2) =m2+m(t) sinkx; .Knowing the density matrix, we can evaluate theratio G(t) = m(t)/m(0):

whereg;(sinkx;) Tr(I;2)

(Al)f «jII& i$Igz=e Igze

Substituting S and K into Eq. (Al), and expandingthe exponentials, we can readily show that

G(f) =5 G„(t)N~p

=Q (E„(f)cosM&d, f+0„(t)sinMu, f), (A2)N~p

where we have

( ) g g;,sinkx; sinkx, 2f "Tr 1 [X„A,'"&]„[K„A~("&]„)n 0 g;(sinkx() Tr(I;2)(2n)! (Ku, )2"

g(~sinkx;sinkx, 2t2 'Tr( [Z„A&s& ],[~„A~& "&]„)g;(sinkx;) Tr(I~2) (2n —1)!(!f&r&,}2

and the constant 5& equals 1 for M & 0 and 5p = 2.To the lowest order in (d„ the dominant &,' ' havethe forms

A~P =I~z(p)

rrr

(M

(A3)

where ~2 and M4 are constants proportional to thecoefficients of t and t' in the series expansion ofE() (t) and have the form

[I X]e

A (-1)[A (1&]1

A (-2) [A (2 & ]t

The preceding expression of G(t) is valid onlywhen &d, is appreciably greater than ( (&(d)„,)'For [(«u)„b/&d, ]=+, we find E,(0)/E(&(0)=1/40 andE2(0)/E, (0) & 1/100. Since the value E„(0}equalsthe area under each absorption line and representsthe intensity at each line, we may neglect all G„(t)with M~1 and assume G(t)=E2(f) Using the re.-lation"

- g, x 2,,Tr [X„I,, jx„I,,]g, x'„.Tr[X„[X„I„][X„[H I„]]

where x;;=x; —x, .(A4}

—(2fkd, ) "M2„+TrI;2=2k Z(x; —x,)t $$

x Tr [X„I,,]„[X„I„]„,we obtain the diffusion constant due to the nonsec-ular dipolar interaction in the form D, =K/H„where K is independent of H, and is given by

r +rT (rr„rIrr r„)rT (r„,,)'I.,

«Work supported in part by the National Science Foundationunder Grant No. NSF-GP-14243, and in part by the JointServices Electronics Program under Contract No.DAAB07-69-C-0383.

'A. G. Redfield, Phys. Rev. 98, 1787 (1955).'A. Abragam, The Principles of Nuclear Magnetism (Oxford U.

P.,London, 1961).'D. Tse and S. R. Hartmann, Phys. Rev. Lett. 21, 511 (1968).

'I. J. Lowe and D. Tse, Phys. Rev. 166, 279 (1968).'H. E. Rorschach, Jr., Physica (Utr. ) 30, 38 (1964).T. Forster, Z. Naturforsch. A 4, 321 (1949).

'The critical radius in Ref. 3 was defined by( ptp ) ( )

1 + 3 cos'8 pr '),f2= H d . The plus sign should be

changed to a minus sign.'Allowance must be made for the typographical error in Ref. 3

which gave g = 3m"'N, C'"l2 instead of the correct

NUC LEAR SPIN- LATTICE RE LAXATION IN. . . 4085

g = 277 N (, ~2/3P

'A. G. Redfield, Phys. Rev. 116, 315 (1959).' J. H. Van Vleck, Phys. Rev. 74, 1168 (1948)."M. Lee and W. I. Goldburg, Phys. Rev. 140, A1261 (1965).' D. Tse and I. J. Lowe, Phys. Rev. 166, 292 (1968).

' T. L Guzzle and P. P. Mahendroo, Phys. Rev. 150, 361(1966).

'4A G Redfield and W N Yu Phys Rev 169 443 (1967)"V. V. Osiko, Fiz. Tverd. Tela 7, 1294 (1965) [Spv.

Phys. -Solid State 7, 1047 (1965).