theory of cross relaxation zobov provotorov sov phys solid state 1975 17 5 839

7/22/2019 Theory of Cross Relaxation Zobov Provotorov Sov Phys Solid State 1975 17 5 839

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of cross relaxationV. E. Zobov and V. N. Provotorovtn*trute of chemical Phpics, Academy of sciences of the ussR, Moscow(Submltted November 1, 19?4)Fiz. Tverd. Tela, 1?, 1298-1304 (May 19?5)The probability of two-spin cross relaxation is calculated with allowance for gradual rransfer of energyfrom dre two. flipPed spins to the remaining spins of the system. The rare of loss of energy by the paii,vhich depends on the way in which the field of the external spins varies. is esdmated io the framework of*te random-field method for discontinuous and continuous variations of rhis field. It is found that in rhelatter case the flow of energy from the pair to the other spins of rhe sysrem is slower.

h tbe physics of the magnetic resonance of solids an im-portatt role is played by cross-relaxation processes,rhtch were first considered in ref. 1. In that paper theprobability of cross-relaxation processes was calculated11 accordance with the known expression of perturbationtbeory for systems with a continuous energy spectrum.for the simplest two-spin cross-relaxation transition, insNch two spins i and j of different species with resonancefrequencies @ro and c,r2s are simultaneously flipped into theopposite direction (flip-flop process), this expression is

urj -h-2 lV;i It g (*1:), (1)*here Vi3 is the matrix element of the transition; g(ro) isthe level density of the continuous spectrum in the finalEtete, a form ftmction that reflects the ability of the spinsystem to tiberate or take up the energ'y firl, = h(arro-oro)'

Then, in refs. 2-4, kinetic equations were derivedqusntum-statistically, these describing the behavior of atpin system with allowance for cross-rela:


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Horvitzts result for the cross-relaxation probabilitya pair of spins when there are discontinuous changes

;of tbe frequency difference between the values co!, +Awltb mean frequency R of the jumps has the form (in ournotation)

termined by the minimal root (-p*in) of the denominatorof (15b). When t >>R, we havewhere

("r (t) - cr (t)) = (o, (0) - c, (0)) exp (-p,rot), (17)

pnro * tu?2R IBi- B,o- @il" * Btrol, (18)in which B'1 and Br2 8r the values of 81 and 82 for s = R,and B1s and 826 are the values for s = R and q(Arz) =d(Arz).

To obtain a concrete result, we take #tl in theform of a rectangle with half-width A = THloc:

(13)

The result (13) has been obtained by means of per-turbation theory, although for a model of motion as simpleas uncorrelated jumps of the frequency difference the prob-lem of the motion of a pair of spins can be solved rigor-ously for both large and small values of o.r!2, even whenperturbation theory cannot be used.

For suppose the frequency difference [or rather thepart Ar2(t)l varies discontinuously, its value after the jumpA12 being unconelated to its value A", before the jump,and that the probability that A.r2 is realized is determinedby the relative fraction of states with such frequency dif-ference g(Arz).

The distribution function q(Arz) is determined by thereal positions of the external spins, and therefore itsform can be readily determined qualitatively in any con-crete case. This must be appreciably nonzero only overthe frequency interval A yHloc. To specify the timevariation of a random variable one must also introducethe mean frequency R of jumps, which in order of magni-tude must be equal to the proba"bility of spin-spin orfl.ip of the spins that produce the dipole fieldthe pair.Since the equations of motion (6) of tbe pair and the

equations of motion of a magnetic moment are the same,problem now reduces to that of the motion of a spin" discontinuously varying longitudinal field, and it cansolved by tbe appropriate methods.ls't{ [r refs. 13 and14 the stationary solution of the analogous problem wasi.considered. We are interested in the nonstationary be-bavior. Performing calculations similar to those maderef. 13, for the Laplace transform of the difference ofthe polarizations averaged over all teaLlzatilons of the ran-:dom process we obtain [for 612(0) = 0, o12(0) = 0]

or (p) - u: (p) : j "-rr,", (r) -or gl> tt -(cr (0) -', (0)) ff;i, (14)F):lr - 81(Rr+4"ttl [s -.R8, (s'* 4o?Jl + BZR(sl? * 4air), (15a]

t19b)-[p * arR (tuiz- pr]l {s - BrR (tulz* s')l + B2$"?r- sp) Rt,(15b)whichs=P+R,and81 :-jBr-

t (.)t2 { 4oi, * ('lg *.)t

The inverse Ia.place transformation for (14) cannotperformed, although fOr c,l!, )>L, atz one can find the

Restricting ourselves in the expansion of B11 and B'2in the small quantity L2/@1i2 to the largest terms, weobtain from (18)T2A2Rwn:| Prarn t T "it iff.

hl, Atfttorr:ifif]i@t@.

9 (ro) (ro!s f ro)qe:ffiqftjrd,'.

? (ltr) : *, .lg-A(Ar:(.lz*A0, Ar, ) .l: * A, Atr (.?r - J.

tu,-lctI-o

It can be seen from (17) that during the motion of thepair the polarization difference changes appreciably overa time [p*io]-t, i.., pmin determines the cross-relaxa-tion probability w12.Comparison of (19) and (13) shows that the asymptoticfrihaviorJglz tta, a12')-ef the rigorous solution leads tothe same rbsult as perturbbtion theory lthe difference inthe numerical coefficient is due to the different choice ofq(A12) l.We now consider a different interesting case: olz =0. For symmetric distribution firnctions, according to(16), 82 = 0 and (14) becomes

or(p) -er(p):ffi (zr(0)-ar(0)). (20)In the simplest case g(Atzl = L/Z[d(Arz + A) + 6(4rz-A) I we havecr (p) - or (p) :tO=n#% (41 (0)-c2(0))- (21)

The roots pt, p2, pr of the third-degree pollmomial inthe denominator in (21) can be calculated by, for example,Cardan formulas. Then the variation in time of the polar-ization will be girren by the e:presslon

dr (r) - q (t): (dr (0) - c1 (0)) [creP,' * "ret't * "rrPJ]. (221In general, cumbersome expressions are obtained forthe roots, so that we give the result for two limiting cases,restricting ourselves in (22) to the terms with largest,r.^\ amplitude and retaining the largest terms in the e4pansion[ro' in the arguments of the exponentials.For Ra12 >t A2(R I anl we have

(o, (r) - ar (t)) = (a1 (0) - or (0)) "rp {- & lrlTxco"{2,,t(t.*ft')}.

(1e)

vior of (ar(t) - oz(t)) for large times, this being de- (23)


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For Rar, afl to the value -orz(0), then p (seewhich always follows from H(see Eq. S), flips, and thimeans that both spins flip, Le.,such proc",:- ;;l;;';;l?, ""," in rhespin dlmamics; for if some spin flips sufficiently slowits flipping will be accompanled, in accordance with thadiabatic theorem, by the flipping of two other spins tinteract with one another, one of them being next to thflipped spin.

\at-b"-b.rg*, S. Shapiro, P. S. Pershan, and J. O. Artman, Phys. RLL4, 445 (1959).'ili. Prouotorov, Zh. Eksp. Teor. Fiz., 42,882 (1962) [Sbv. Php. - Jr5,611 (1962)1.3f L. Suirhuili. lr.1. D. Zviad,adze, and G. R. Khutsishvili. Zh. Eksp. TeFiz., &,8?6 (1968) [Sov. Phys. - JETP, 27, 469 (1968)].aL. t. Eiishvili and N. P. Giorgadze, teoi]Ilar. Fiz., L2, 420 (19'i2).fu. fctr. Kopviltem, Fiz. Tverd. TeIa,2, f 829 (1960) [6v. Phys. - SolState,2,1653 (1961)1.6tt. Hirino, J. Phys. Soc. Japan, 16, ?66 (1961h l?, ?88 $962).tW. J. C. Granr, phys. Rev., rSS. IZ:,5S ftg64).tw. J. cr"nr. Phys. Rev., 134]f354 (1964,; 134, 1565 (1964).sA. Abr:rgam, The Princiffi of Nuclear trta@tism, Oxford (1961), ChII and VIII.roJ. R. Kaluder and P. W. Anderson, Phys. Rev., 125, 912 (1962).rlA. D. \,lilov, K. lvt. Salikhov, and Yu. D. Tsvetkov. Zh. Eksp. Teor...63, 2329 (19lQ)-_[Sov. phys. - JETp, 36, 1229 (i9?3.1.l-'tp. Horvirz, liiiy"-a"u. (B,r, 3, 286? tr9?lr.r3A. I. Burshtein. Zh. Eksp. teo]. riz., 54, 1120 (1968) [Sov. phys. - J21. 600 (1968)1.1{ l,t. Salikhov and A. B. Doktorov, in: Paramagneric Resonance , Lg1969 lin RussianJ, Kazan (f 970), 4.2, p. L22.ltc. W. Parker. Amer. J. Phys.. 38, 1432 (19?0).rtF. L. Aukhadeev, I. I. Valeev, L S. Konov. V. A. Skrebnev. and \1. ATeplov, Fiz. Tverd. Tela, 15, 235 (19?3, [Sov. Phys. -Solid Srate, 1163 (19?3)1.l7O. Takashi, K. Tatsuo, T. Toshihiko, and S. \litsuo, Solid Stare Commun., 13, 643 (19?3).lEvon R.?remer, Phys. Status Solidi , 42, 507 (19?0).reY. Tokunaga. S. Ikeda, K. Ito, and ilHaseda, J. Phys. Soc. Japan,351353 (r9?3).zoF. 81och, Phys. Rev., ?0, 460 (1946).

(241We now return once more to the base of large fre_quency differerices, c,{z >> yHloc. Comparing (1g) and (12)with (1) and noting that for the flip-flop process lVrrl, =al2, we obtain for the considered nonresonance cross-re-la:cation mechanism g(c.,!2) x znzR/J(rh)4 for a discontinu-ous change in the frequency difference and g(c,rl2) * Z(i\).tT3&rl/61-1 for a slow continuous variatiou.As a rrrle, the resonance mechanismlrs gives astronger dependence on co!2, since targe dipole fi.elds areneeded to compensate large frequency differences. Srch .fields can be formed for a definite arrangement (orienta-tion) of a large number n - rlr/yH1os of spins in the en-vironment of any pail, but the probability of this arrange-ment is proportionalls to exlp{ -n2}, and therefore in theresonurnce mechanism the probabi.lity of cross relaxationbetween two lines must decrease with increasing frequencydifferenc" ,1, between these lines as exp {-(,':lz-/vHlod2}.With allowance for the nonresonance cross-relaxationmechanism, which also occurs in homogeneous systemsconsisting of spins of two species, the form of the asymp-

totic behavior is different. Going over from the probabil-ity of cross relaxation between two spins wii (1), (19) , (LZlto the probability of cross rela:ration betweeh two linesW12 (this transition can be madet-8 by summation over thepairs), we obtain

w,,:#) oi,et l,l.i, i .When olz - yHloc, the frequency difference can beeasily compensated by dipole fields, i.e., the resonaaEb \mechanism that proceeds through resonance pairs willpredominate in the cross relaxation. Conversely, when,\z>r yHloc, there are few resonance pairs and the non-

re sonance cro ss -relaxation mechani sm, which occur s i nall pairs and gives a power dependence of W,2 on the fre-quency difference, will predominate.It is interesting to note that in some experimeng"l6-19at large frequency differences a transition from a Gaussiandependence of the cross-relaxationprobability on the fre-quenc)' difference to a weaker, possibly power dependencewas noted.In conclusion we return once more to Eqs. (6) and

t

U2 So,. Phyu Solid State, VoL 1?, No 5 V. E. Zoborr and V. N. Provotoror

theory of cross relaxation zobov provotorov sov phys solid state 1975 17 5 839

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