Phase relaxation in a many-body system of diffusing spins: Slow motional limit

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 1 1 JULY 22 Phase relaxaion in a many-body sysem of diffusing spins: Slow moional limi Alexander A. Nevzorov a) and Jack H. Freed b) Deparmen of Chemisry and Chemical Biology, Cornell Universiy, Ihaca, New York Received 7 March 22; acceped 5 April 22 The echo ampliude decay of a diffusing elecron-spin-bearing molecule ineracing wih a diffusing many-spin bah of proon-conaining molecules has been sudied heoreically in he slow moional limi. Closed-form asympoic expressions for he shor- and long-ime behavior of he echo envelope have been obained. In conras o he well-sudied fas-moional limi, he echo envelope canno be described by a simple exponenial decay, and is rae exhibis a D T 1/3 dependence on he relaive ranslaional diffusion coefficien, D T. For low proon concenraions and long pulse delay imes,, an exponenial 9/8 -decay of he echo ampliude is found, whereas for high proon concenraions and shor s he decay exhibis an exponenial 3 -behavior. These limiing analyical resuls are compared wih he exac numerical soluions o esablish heir range of validiy. 22 American Insiue of Physics. DOI: 1.163/ I. INTRODUCTION In he fas moional limi, he classic heories of spin relaxaion used in NMR and ESR in fluid media have been available for many years. 1 These heories are formulaed for he roaional diffusion of a single spin-bearing molecule or else he relaive ranslaional diffusion of a pair of spinbearing molecules. These approaches are perurbaional in naure, based on he rapid sochasic modulaion of he dipolar and oher ineracion ensors by he moion. In viscous media slow moions he convenional moional narrowing heories are no longer valid, and a full soluion o he sochasic modulaion of he spin dependen ineracions mus be employed. The usual slow moional spin relaxaion analyses for hese cases are based on he sochasic Liouville equaion SLE. 2 Their relevance is well esablished for a wide range of ESR and also NMR experimens. 2,3 However, hese one- or wo-body moional modulaion cases do no adequaely address he case of many spin-bearing molecules simulaneously ineracing by heir dipolar ineracions, while hey are separaely diffusing. In he moional narrowing or Redfield regime, Torrey developed a successful ad hoc model, 4 which is appropriae for dilue soluions. In his model, he many-body effecs were aken ino accoun by a simple muliplicaion of he inverse relaxaion imes T 1 1 or T 1 2 for a pair of ineracing spins by he number densiy. However, when he moions slow down sufficienly, hen he many-body ineracions persis for long imes and he Torrey approach does no generalize o he slow moional regime. Recenly, we have developed a many-body heory for spin relaxaion o address his general issue. We showed ha under he assumpion of saisical independence of he moions, i is possible o express he ime evoluion of he a Presen address: Deparmen of Chemisry and Biochemisry, Universiy of California San Diego, 95 Gilman Drive, La Jolla, CA b Auhor o whom correspondence should be addressed. Elecronic mail: jhf@ccmr.cornell.edu many-body free inducion decay or echo decay as a general funcion of he wo-body soluion, 5 cf. below, and he laer is solved using he SLE. 6 The numerical resuls obained recover he Torrey Redfield resuls in he moional narrowing regime and also give he correc resul, known as Anderson s saisical heory 1,7,8 in he limi of a dilue solid, as well as a full descripion of he inermediae regime beween hese wo limis. In one of our recen conribuions, 9 wo channels for he relaxaion of an elecron- spin in a many-spin bah of proons have been sudied. The firs is spin diffusion, which predominanly governs he relaxaion in he rigid limi. This yields an echo ampliude decay ha is quadraic in ime in he exponenial due o he presence of flip-flop erms in he homonuclear proon par of he Hamilonian. 9 When moions occur, a dephasing arises, which is due o he ranslaional diffusion of he proons relaive o ha of he elecron spin. This laer dephasing becomes comparable wih spin diffusion even for ulraslow moional raes, i.e., when he diffusion coefficien for relaive moion is less han D T 1 12 cm 2 /s. The echo envelope due o such a diffusional dephasing has been sudied so far only numerically. 5,9,1 Wha has been lacking is an analyical closed-form resul appropriae for he slow moional regime o provide he insigh no readily obainable from he numerical soluions. I is he purpose of his aricle o repor on his analyic soluion appropriae for very slow moions and o demonsrae is range of validiy by comparison wih he full numerical resuls. We describe he new insighs provided wih respec o he nonexponenial decay of he phase memory in spin echo experimens. These resuls are especially relevan o ESR experimens in viscous media such as liquid crysals and glass forming fluids, 5,9,11,12 as well as NMR in polymers and solids. 13,14 The case ha we explicily consider is relaxaion of a single spin wih a large magneic momen e.g., an elecron /22/117(1)/282/6/$ American Insiue of Physics

2 J. Chem. Phys., Vol. 117, No. 1, 1 July 22 Phase relaxaion in a sysem of diffusing spins spin in he field of many diffusing spins wih weak magneic momens e.g., proons. For his case we are able o sar wih he fundamenal general resul previously derived cf. Eq ,9 This general resul also applies o anoher case, ha of many idenical spins, each diffusing independenly. Thus he presen approach also applies o his second case, bu i is complicaed by oher facors viz. he generaion of muli-quanum coherences, 11 which can in principle be suppressed by using magic echo echniques. Thus our resuls are presened in he conex of he firs case. II. ENSEMBLE-AVERAGED SOLUTION TO THE ECHO SIGNAL IN THE PRESENCE OF MOTIONS We sar by employing he expression for he echo signal, G() of an elecron spin S ineracing wih a dynamical bah of N idenical proons I (i), which is obained from solving he von Neumann equaion for he many-spin densiy marix in Liouville or superoperaor form: 5,9 G Z gt i e d2 H x ( 2 ) i Xx O e d1 H x ( 1) O g g T e O i d2 H x ( 2 ) eo i d1 H x ( 1 ) X x g, 2.1 which is Eq. 5.5 in Ref. 9. Also 15 H x()x x H x ()X 1 x.in Eq. 2.1 he vecor g() describes he many-spin densiy sae immediaely afer he iniial (/2) x pulse. I corresponds o a represenaion of he densiy marix in Liouville space, which is defined by he Frobenius race meric cf. Eq. 2.8 in Ref. 5. H x and X are he roaing frame Hamilonian and pulse superoperaors, respecively, Z is a normalizaion consan, and he angular brackes imply ensemble averaging. The spin Hamilonian ha is appropriae for his slowmoional case of a single A elecron spin ineracing wih N spins of ype B proons is 9 N1 H i2 F r 1i S z I z (i) 1 F r 1i S z I (i) F r 1i S z I (i). 2.2 As usual, he nonsecular elecron spin-flip erms have been negleced in Eq. 2.2, since hey are no imporan, excep for very fas moions. 2 Whereas he pseudo-secular S z I erms are, in general, significan, hey were found in he numerical soluions o be unimporan for he very slow moional range over which he asympoic soluions o be obained in Sec. III are valid excep for very weak echoenvelope modulaion. 9 Thus we shall neglec hem below. We shall also neglec he addiional relaxaion by spin diffusion, or assume ha i is suppressed by oher means e.g., he moional averaging 9 or by proon spin-locking a he magic angle. Now if he moions of he spins are assumed o be sochasically independen, he echo signal is given by 5,9 cf. he closely relaed case given by Eqs of Ref. 5 exp i 2 d1 s 1 Fr 1 exp i 2 d1 s 1 Fr 1 N, 2.3 G2 (N1) q where q/kt, (16/5) A B is he coupling consan, and he r-dependen funcions conaining he implici ime dependence are given in erms of he spherical harmonic of rank wo, F(r)F (r)y (2) (,)/r 3. The s-funcion is given by s()1,, and s()1,. In he hermodynamic limi of a very large number of spins, N, and a large volume, V, he Markov mehod 16 can be applied o Eq. 2.3, which yields 5,9 GGexp C d1 g 1 1, 2.4 where CN/V is he concenraion of he marix proons in he presen case. Also, g()g ()g ()/2 and he wobody spin-echo signal from he S spin ineracing wih a single I spin is given by g 1 exp i 2 1d2 s 2 Fr In Eq. 2.5, he prime means ha he volume has already been facored ou. 5 The moionally pah-averaged componens g () are found from he soluion of he SLE 2 for he auxiliary funcion g (r,): g r, D T 2 g r,i 2 Frsg r,, 2.6 where we have g ()d 3 r g (r,). Here D T is he relaive ranslaional diffusion coefficien for a pair of ineracing spin-bearing molecules i.e., a proon-bearing and an elecron-spin-bearing molecule, which is given by he sum of heir respecive diffusion coefficiens. The iniial condiion is given by g (r, )p eq (r), he equilibrium probabiliy disribuion. For he wo-body echo signal, g (), one can formally wrie he soluion of Eqs. 2.5 and 2.6 as g d 3 r e [i(/2)f(r)d T 2 ]() e [i(/2)f(r)d T 2 ] p eq r. 2.7 In summary, we have, under he assumpion of sochasic independence of he diffusion of he spin-bearing molecules, ha he echo decay in he many-body case is an analyic funcion of ha for he wo-body problem. III. ASYMPTOTIC EXPANSIONS OF THE ECHO BEHAVIOR 283 We now seek asympoic expressions ha are valid in he limi of very slow moions. The oher limi of moional narrowing can readily be obained from Eq. 2.3 as discussed previously 5,9 by uilizing generalized cumulan expansions

3 284 J. Chem. Phys., Vol. 117, No. 1, 1 July 22 A. A. Nevzorov and J. H. Freed and runcaions afer he second order. This approach does no yield a saisfacory mehod for he slow moional limi, so we mus proceed in a differen manner. For 2 we firs use he following exac propery of noncommuing symmeric operaors A and B cf. he Appendix: e (AB) e (AB) e O d exp(a)b exp(a)eo d exp(a)b exp(a), 3.1 where he symbol O (O ) sands for he posiive negaive Dyson ime ordering. This enables one o ransform Eq. 2.7 o g 2 d 3 r g r,2 d 3 r e O e O D d exp[i(/2)f(r)] 2 exp[i(/2)f(r)] T D d exp[i(/2)f(r)] 2 exp[i(/2)f(r)] T peq r. 3.2 For he magneic dipolar poenial, one has from Maxwell s equaions: 2 F(r). Using he well-known Campbell Hausdorff expansion, 17 one can expand he operaor in he ordered exponenials of Eq. 3.2 in a series of commuaors. Forunaely, he hird and higher commuaors vanish, and one obains exacly ha D d 2 (i/2)[f(r), 2 ](1/2)(i/2) 2 [F(r),[F(r), 2 ]] e T O e O D d[ 2 if(r) ( 2 2 /4)F(r) 2 ] T. 3.3 The squared modulus of he dipolar field gradien is readily found o be Fr cos2 1 r r 5 8 cos4 2cos The decay of he elecron spin echo due o ineracions wih he solven proons ha are modulaed by he diffusion is hen given, from Eq. 2.4, by G2Gexp C d r 3 1 g r,2g r,2 2 p eq r, 3.5 where he prime has he same meaning as in Eq Equaion 3.5 is he exac formal soluion o Eqs In he general case i remains a dauning ask o solve, and he mehods of Refs. 5, 6, 9, and 11, yielding accurae numerical soluions, are more useful. Equaion 3.5, however, provides he basis for sudying he slow moional limi i.e., he limi as D T, especially given is linear dependence upon D T in he ordered exponen. Useful forms can indeed be obained from Eq. 3.5 in cerain cases. Firs we noe ha in he absence of any poenial of mean force, 18 one has p eq (r)1. Then one finds ha he erm involving 2 in he exponenial operaor of g(r,2), Eq. 3.2 as given by Eq. 3.3, governs boh he shor- and long-ime behavior of Eq Tha is, a shor imes, one expands he imeordered exponenial in Eq. 3.3 o firs order in D T and only he erm in 2 survives afer posmuliplicaion by p eq (r) 1, so one obains G2Gexp 8 D 5 T d Cd3 A B 3 A B, d where d is he disance of minimal approach beween he elecron spin and he proons. For long, he 2 erm in Eq. 3.3 clearly dominaes, so we drop he oher wo erms. The resuling inegral in he exponen leads o an incomplee gamma funcion which may be evaluaed asympoically 19 in he limi of large or more rigorously D T 2 3 d 8 o yield G2Gexp 4 3 I 8 5 Cd3 3D T d 2 A B 3/8 A B d 3 9/ Noe ha in Eqs. 3.6 and 3.7 all quaniies in he parenheses are combined o form dimensionless facors. In deriving Eq. 3.7 we have used he following inegrals: 1/d 8 (1 e x )x 11/8 dx 8 3 3/8 ( 5 8)d 3 valid for d and I 1 (5x 4 2x 2 1) 3/8 dx1.81. To obain he firs inegral, we have expanded he incomplee Gamma funcion up o he firs order 19 o ensure proper dimensional regularizaion of all parameers in Eq. 3.7 when d. Noe ha since he dipolar-field gradien, F(r), is jus an ordinary funcion of r, he Dyson ime ordering can be omied in evaluaing Eq. 3.5 for he above limiing cases. IV. COMPARISON WITH NUMERICAL SOLUTIONS AND DISCUSSION As can be seen from he superexponenial characer of Eq. 3.5, when concenraions are high, i.e., Cd 3 1, so he echo decay is relaively rapid, hen he shor-ime behavior on he wo-body ime scale governs many-body relaxaion, i.e., Eq. 3.6 will be appropriae. A lower concenraions he phase memory of a many-spin sysem is described by a more complicaed funcion, involving boh Eqs. 3.6 and 3.7, as we show below. The shor-ime 3 behavior exhibied by Eq. 3.6 is o be expeced. I represens he well-known shor-ime resul for single-body relaxaion for such cases as spin dephasing due o ranslaional diffusion in a field-gradien in he Carr Purcell pulse sequence 2 and for modulaion of an an-

4 J. Chem. Phys., Vol. 117, No. 1, 1 July 22 Phase relaxaion in a sysem of diffusing spins 285 FIG. 1. Elecron spin-echo envelopes as a funcion of pulse delay ime a differen proon concenraions in he slow moional regime, D T 1 4 A B /d. The shor-ime expression, Eq. 3.6, and long-ime expression, Eq. 3.7, are represened by dashed and do-dashed lines, respecively; he solid lines are he exac numerical resuls obained from Eq isoropic spin Hamilonian by molecular roaional diffusion. 1 The fac ha he wo-body expression is exponeniaed in Eq. 3.5 will lead o Eq. 3.6 being valid over a longer ime-scale han he wo-body resul, as seen in he comparison wih he exac resuls below. The long-ime 9/8 exponenial behavior of Eq. 3.7 for slow moion is modified from he well-known exponenial behavior in /T 2 in he moional-narrowing regime, as is he power law dependence on D T. Plos of he echo envelopes are shown in Fig. 1 for D T 1 4 A B /d, corresponding o he nearly rigid limi. Case a corresponds o low concenraions, wih inermediae concenraions b, and high concenraions c and d. Solid lines show he resuls of numerical soluion of Eq. 2.3 by using he SLE, Eq. 2.6, cf. Refs. 5 and 9, and he dashed lines correspond o he limiing behavior given by Eqs. 3.6 and 3.7. If he disance of minimal approach is se o d3 Å,a corresponds o raher dilue soluions wih proon concenraions, C of cm 3 ; b is for C cm 3 ; and c corresponds o a C ( cm 3 ) similar o pure waer cm Case d is for unphysically high concenraions of cm 3 o show he limiing cubic behavior of he echo envelope decay. Case a is dominaed by he long ime 9/8 behavior, bu wih 3 behavior a very early imes. Case b shows an increased decreased role for he laer former. Figure 2 shows he echo envelopes calculaed for differen diffusion raes D T, for Cd As in Fig. 1, solid lines designae he numerical soluions. Equaions 3.6 and 3.7 are seen o be useful asympoic expressions for he echo envelopes over he slow moional range, i.e., for D T 1 3 A B /d. Bu he validiy of Eqs. 3.6 and 3.7 breaks down a sufficienly fas moions, i.e., for D T 1 2 A B /d. These dimensionless inequaliies become D T 1 11 cm 2 /s and D T 1 1 cm 2 /s, respecively using a d3 Å and he appropriae gyromagneic raios. Tha is, Eq. 3.6 is applicable only for much shorer imes as D T increases. Equaion 3.7 becomes inapplicable as D T increases, mos likely as a resul of he erms in and 1 in he exponen of Eq. 3.3 becoming more imporan during inermediae ime periods recall ha hese erms in Eq. 3.3 require inegraion over. These comparisons hus demonsrae he essenial validiy of he asympoic forms in he very slow moional regime. In addiion, hey show ha he echo envelope decay canno be expressed by a simple exponenial even a longer delay imes. This explains an apparen anomaly previously seen in analyzing he numerical resuls for he concenraion dependence of he echo decay. This was performed using a simple

5 286 J. Chem. Phys., Vol. 117, No. 1, 1 July 22 A. A. Nevzorov and J. H. Freed FIG. 2. Elecron spin-echo envelopes as a funcion of pulse delay ime a differen ranslaional diffusion raes, D T, as shown. The proon concenraion is se o Cd 3.24 inermediae concenraion regime o capure boh he shor- and long-ime limiing behavior, cf. Eqs. 3.6 and 3.7. The differen lines are as in Fig. 1. exponenial in ime and resuls in a C a wih 5 a.9 which is close o 8 9. This is equivalen o linear dependence on C when he proper 9/8 power is used. Noe also ha he D T 3/8 power law dependence on he diffusion coefficien would become an effecive D T 1/3 power law from fiing he echo envelopes o a simple exponenial decay funcion insead of 9/8, and his is in agreemen wih he.34.2 power law obained numerically. 5,9 The closed-form soluions derived herein break down when moions become sufficienly fas, i.e., when he line broadening is essenially homogeneous. 5 In his limi he well-known Torrey Redfield heory is appropriae e.g., he specral densiy J() ha conribues o he ransverse spin relaxaion is given by J()(4/15)(C/d D T ). In he ulraslow moional regime considered herein, he asympoic equaions for he ransverse spin relaxaion, while more complex, can sill be wrien in erms of he coefficien for relaive diffusion, D T, he disance of closes approach, d, and he concenraion, C. This resul could poenially be of use o aid in sudying ulraslow moional relaxaion in polymers, glasses, and semi-ordered media. One cavea, however, is ha in he rigid limi, relaxaion due o ranslaional diffusion is no longer relevan, and he phase memory decay for he elecron-spin is dominaed by effecs of nuclear spin diffusion, which, however are readily averaged ou by moion. 9 ACKNOWLEDGMENTS This work was suppored by Grans from he NSF and NIH/NCRR. APPENDIX: DERIVATION OF EQ. 3.1 To derive Eq. 3.1 we noe ha he ime evoluion of an arbirary funcion g() obeying he equaion g ABg A1 is given by g()exp(ab)g(). By subsiuing g() exp(a)ĝ() ino Eq. A1 and inegraing we find ha e (AB) e A e O d exp(a)b exp(a). Or, by aking he ranspose of boh sides of Eq. A2, e (AB) e O d exp(a)b exp(a)e A A2 A3 since A and B are symmeric. If we replace A by A in Eq. A2 and muliply i by Eq. A3 from he lef we ge Eq. 3.1.

6 J. Chem. Phys., Vol. 117, No. 1, 1 July 22 Phase relaxaion in a sysem of diffusing spins A. Abragam, The Principles of Nuclear Magneism Oxford Universiy Press, London, D. J. Schneider and J. H. Freed, Adv. Chem. Phys. 73, P. P. Borba, A. J. Cosa-Filho, K. A. Earle, J. K. Moscicki, and J. H. Freed, Science 291, H. C. Torrey, Phys. Rev. 92, A. A. Nevzorov and J. H. Freed, J. Chem. Phys. 112, A. A. Nevzorov and J. H. Freed, J. Chem. Phys. 112, P. W. Anderson, Phys. Rev. 82, J. R. Klauder and P. W. Anderson, Phys. Rev. 125, A. A. Nevzorov and J. H. Freed, J. Chem. Phys. 115, L. J. Schwarz, A. E. Sillman, and J. H. Freed, J. Chem. Phys. 77, A. A. Nevzorov and J. H. Freed, J. Chem. Phys. 115, J. H. Freed, Annu. Rev. Phys. Chem. 51, V. J. McBriery and K. J. Packer, Nuclear Magneic Resonance in Solid Polymers Cambridge Universiy Press, New York, F. A. Bovey and P. A. Mirau, NMR of Polymers Academic, New York, X In Eq. 3.3 of Ref. 9 he ilde over H AB is missing in he second erm. In addiion, he sign of he commuaor has been accidenally swiched in Eqs This, however, does no affec he final resul, Eq of Ref. 9, which does no depend on he sign, i.e., he decay due o spin diffusion is independen of he sign of he gyromagneic raio for he elecron spin. 16 S. Chandrasekhar, Rev. Mod. Phys. 15, E. Merzbacher, Quanum Mechanics, 2nd ed. Wiley, New York, L. P. Hwang and J. H. Freed, J. Chem. Phys. 63, Handbook of Mahemaical Funcions wih Formulas, Graphs, and Mahemaical Tables, edied by M. Abramowiz and I. Segun U.S.P.O., Washingon, DC, C. P. Slicher, Principles of Magneic Resonance, 3rd ed. Springer-Verlag, Heidelberg, Noe ha our analysis neglecs he correlaed relaxaion effecs of he pair of proons on each waer molecule.