Spin relaxation of radicals in low and zero magnetic field

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 8, NUMBER JANUARY 003 Spin relaxation of radials in low and zero magneti field M. V. Fedin International Tomography Center SB RAS, Institutskaya st. 3a, Novosibirsk, , Russia P. A. Purtov Institute of Chemial Kinetis and Combustion SB RAS, Institutskaya st. 3, Novosibirsk, , Russia E. G. Bagryanskaya a) International Tomography Center SB RAS, Institutskaya st. 3a, Novosibirsk, , Russia Reeived July 00; aepted Otober 00 Spin relaxation of radials in solution in low and zero magneti field has been studied theoretially. The main relaxation mehanisms in low magneti field modulation of anisotropi and isotropi hyperfine interation, and modulation of spin rotational interation are onsidered within a Redfield theory. The analytial results for a radial with one magneti nuleus (I ) and for a radial with two equivalent magneti nulei (I ) are obtained and analyzed. It is shown that the probabilities of relaxational transitions in low and zero magneti fields differ signifiantly from the probabilities in high magneti fields. The use of high-field expressions in low and zero magneti fields is not orret. Taking exat aount of spin relaxation is important in alulations of muh low-field magneti resonane data. 003 Amerian Institute of Physis. DOI: 0.063/.530 I. INTRODUCTION Eletron and nulear spin relaxation of radials in solutions is one of the important fators determining magneti and spin effets suh as hemially indued dynami nulear polarization CIDNP, hemially indued dynami eletron polarization CIDEP, magneti field reation yield MARY, stimulated nulear polarization SNP, et. 7 Spin relaxation in a high magneti field has been well studied both experimentally and theoretially during the last deades. But reently many experimental and theoretial works have been onerned with radial reations in low and zero magneti fields, due to both fundamental interest and possible biologial appliations. 8 Despite the fat that spin relaxation in a low magneti field has been orretly taken into aount numerially in several works e.g., in Ref. 6, to the best of our knowledge, the analytial onsideration with a detailed analysis has not been done so far. The main mehanisms of spin relaxation in a low magneti field are the modulation of an anisotropi hyperfine interation HFI, of isotropi HFI, of spin rotational interation SRI and of eletron spin exhange. In a reent paper we have shown that the rate of spin relaxation indued by anisotropi HFI in low magneti fields is signifiantly different in omparison with a high magneti field, and thus the use of high-field expressions for relaxation times is inorret. 9 The onsideration was done for a radial with one magneti nuleus with spin I. In this paper we extend this researh to radials with a more ompliated HFI struture and for the other relaxation mehanisms. This investigation was onentrated on the relaxation of spin state populations, a Author to whom orrespondene should be addressed. Fax: Eletroni mail: elena@ tomo.ns.ru while the analysis of phase relaxation will be a topi of further researh. II. GENERAL REMARKS Eletron and nulear spin relaxation of radials in solutions is usually alulated using the Redfield relaxation theory. 0 This approah allows one to alulate the probabilities of relaxational transitions between spin states of radials in arbitrary magneti fields. Briefly, the alulation involves the following steps. The spin Hamiltonian of the system is written as Ĥ Ĥ 0 Ĥ t, where Ĥ 0 is the stationary spin-hamiltonian, and Ĥ (t) is the time-dependent stohasti perturbation whose average value is equal to zero. The spetral density of noise indued by Ĥ (t) at frequeny an be alulated as J ij,kl i Ĥ t j l Ĥ t k, where i, j, k, l refer to the stationary states of the system, the line orresponds to the averaging over all possible realizations of Ĥ (t), and is the orrelation time of the stohasti proess. Then, the matrix of spin relaxation an be alulated using the following expression: R ij, kl J ik, jl E j E l J ik, jl E i E k jl n J nk, ni E n E k ik n J nj, nl E n E l, /003/8()/9/0/$ Amerian Institute of Physis Downloaded 03 Jan 003 to Redistribution subjet to AIP liense or opyright, see

2 J. Chem. Phys., Vol. 8, No., January 003 Spin relaxation in low and zero field 93 where E i, E j, E k, E l orrespond to the energies of radial spin levels in frequeny units, ij for i j and 0 for i j. Matrix elements R ij, kl represent the probabilities of relaxational transitions between states of the density matrix ij and kl. Matrix elements R ii, jj an be interpreted as the probabilities of relaxational transitions between spin levels i and j. For most pratial ases it is enough to use diagonal matrix elements R ii, jj only. However, taking orret aount of phase relaxation via elements R ij, kl an lead to some quantitative improvements. It is lear that the matrix of spin relaxation annot be alulated for a radial with an arbitrary number of magneti nulei n, sine the dimension of Rˆ inreases drastially with n. Therefore, we onsider here the two simplest ases of a radial i with only one magneti nuleus n, and ii with two equivalent magneti nulei n. The analysis of results obtained for these two ases allows us to draw general onlusions about the spin relaxation of arbitrary radials in a low and zero magneti field. The stationary Hamiltonian of a radial with an n magneti nuleus is Ĥ 0 e Ŝ z n Î z aŝ z Î z a Ŝ Î Ŝ Î, 4 where e and n are Zeeman eletron and nulear frequenies and a is the isotropi HFI onstant. The eigenfuntions and energy levels an be found from the Breit Rabi expressions, where we neglet n in omparison with e : e n, E e a 4, C e n C e n, E a 4 e a, 5 FIG.. The sheme of the energy levels of a radial with one magneti nuleus I / a and a radial with two equivalent magneti nulei I / b. e n n, E a e, C e n n e n n ) C e n n, E 4 a 9a 4a e 4 e, 3 e n, E 3 e a 4, 3 C 3 e n n e n n ) C 4 e n n, 4 C e n C e n, E 4 a 4 e a, where C (/)( e / e a ), C (/)( e / e a ). The sheme of the energy levels is shown in Fig. a. For a radial with n equivalent HFI onstants a the stationary spin-hamiltonian an be written as Ĥ 0 e Ŝ z n Î z Î z aŝ z Î z Î z E 3 4 a 9a 4a e 4 e, 4 e n n, E 4 a e, 5 C 4 e n n e n n ) C 3 e n n, E 5 4 a 9a 4a e 4 e, 7 a Ŝ Î Î Ŝ Î Î. 6 6 C e n n e n n ) C e n n, Negleting n in omparison with e, one an alulate the eigenfuntions and energy levels of a radial: E 6 4 a 9a 4a e 4 e, Downloaded 03 Jan 003 to Redistribution subjet to AIP liense or opyright, see

3 94 J. Chem. Phys., Vol. 8, No., January 003 Fedin, Purtov, and Bagryanskaya 7 e n n e n n ), 8 e n n e n n ), where C a e 9a 4a e 4 e, E 7 e, E 8 e, Here a iso a is the isotropi HFI onstant, all the other averages F i (t)f j *(t) are equal to zero; and have their usual meanings of azimuthal and polar angles in spherial oordinates. It has been shown in Ref. 9 that in the zero magneti field e 0 relaxation transitions are not allowed between spin state 4 and 3 other degenerate spin states,, 3. In ontrast, in the high magneti field e a all 6 possible transitions are allowed. The matrix of relaxation R for the two limiting ases e 0(R 0 ) and e a (R ) in Liouville basis,,,, 3,3, 4,4 an be written as 9 C a e 9a 4a e 4 e, C 3 a e 9a 4a e 4 e, C 4 a e 9a 4a e 4 e. The sheme of the energy levels is shown in Fig. b. Eigenstates 6 orrespond to the total nulear momentum I ; eigenstates 7 8 orrespond to the total nulear momentum I R A:A, 3 v 3v 3 3 5v 3v v Rˆ 3 5v v 3v 3 5v 3 3v v 3 3 5v 0 A:A, 0 III. SPIN RELAXATION CAUSED BY MODULATION OF ANISOTROPIC HFI The modulation of anisotropi HFI by rotational motion is one of the major relaxation mehanisms of radials in solutions. Anisotropi HFI-indued spin relaxation in a low magneti field for a radial with one magneti nuleus I has been onsidered in Ref. 9. The Hamiltonian for an anisotropi dipole dipole interation between the eletron and nuleus an be written as follows: 4 Ĥ t Ŝ z Î z 4 Ŝ Î Ŝ Î F 0 t Ŝ Î z Ŝ z Î F t Ŝ Î z Ŝ z Î F * t Ŝ Î F t Ŝ Î F * t, 8 where the random funtions F are given by F 0 t q 3os t, F t 3q sin t os t exp i t, F t 3q 4 sin t exp i t, 9 F 0 t F 0 t 4 5 q, F t F * t F t F * t 3 0 q, q 6 A:A 6 i x, y, z a i a iso. FIG.. Magneti field dependene of probabilities of relaxation transitions aused by the modulation of anisotropi HFI for a radial with one magneti nuleus I / a and for a radial with two equivalent magneti nulei I /, b. Calulated urves orrespond to the probabilities of relaxation transitions between spin levels i and j as shown in the figure in the extreme motional narrowing limit. Downloaded 03 Jan 003 to Redistribution subjet to AIP liense or opyright, see

4 J. Chem. Phys., Vol. 8, No., January 003 Spin relaxation in low and zero field 95 where v / e. In the low magneti field e 0, e a the relaxational transitions 4,, 3 are allowed, but their probabilities are very small in omparison with a high field Fig. a. The matrix of anisotropi HFI-indued spin relaxation of a radial I / in an arbitrary magneti field is given in Ref. 9 for the ase a, e ). The derease of the probabilities of relaxation transitions 4,, 3 in a low magneti field leads to some speifi features in magneti and spin effets. 9 For example, the time profiles of the low-field ( 4 ) and high-field ( 3 ) Time- Resolved EPR TREPR kinetis are similar in the high magneti field. But, in a low magneti field the deay of the low-field line is muh slower than that of the high-field one, if the hemial reation is slower than the spin relaxation. This feature has been observed in the experiments on lowfield CIDEP, 5,6 in omplete agreement with the theory. For the low-field taking a orret aount of the anisotropi HFIindued relaxation is also important for the alulations of CIDEP and EPR spetra, field dependene of CIDNP, magneti field effets on reation yield MARY, and other lowfield spin effets. 9 To draw general onlusions on the trends of anisotropi HFI-indued relaxation in the low magneti field for the radials with a more omplex HFI struture, we onsider below the relaxation in a radial with equivalent nulei I /: Ĥ t Ŝ z Î z 4 Ŝ Î Ŝ Î F 0 () t Ŝ Î z Ŝ z Î F () t Ŝ Î z Ŝ z Î F () * t Ŝ Î F () t Ŝ Î F () * t Ŝ z Î z 4 Ŝ Î Ŝ Î F 0 () t Ŝ Î z Ŝ z Î F () t Ŝ Î z Ŝ z Î F () * t Ŝ Î F () t Ŝ Î F () * t, where the random funtions F are given by the same expressions as in 9, F i () F i (, ), F i () F i (, ), supersripts and subsripts and orrespond to the first and the seond nuleus. In addition to 9, one needs to take into aount the following: F 0 () t F 0 () t 5 q 3 os 0, F () t F () * t F () * t F () t F () t F () * t F () * t F () t 3 0 q os 0 os 0, where 0 is the angle between radius-vetors from an eletron to eah nuleus; all the other averages F i () (t)f j () *(t) are equal to zero. Thus, the harater of spin relaxation of a radial with equivalent nulei depends on geometry of a radial through the parameter 0. Following the steps desribed in Se. II, we have alulated the matrix of relaxation R. We present here the result for two limiting ases e 0(R 0 ) and e a (R ) only, for brevity, assuming a, e. In the Liouville basis of spin eigenstates,,,,..., 8,8 : 8 s 8 s 0 4 s s 3s s 8 s 5 s 3s 0 8 s 3 s 4 3s 9s 9s 8 s 0 5 s 3s 8 s 4 3s 3 s 9s 9s 0 8 s 8 s 6 7 s s 4 s s 3s Rˆ 6 7 s 0 4 s 3 s 4 3s s 0 8s 3s s 4 3s 3 s 4 s 0 0 8s 3s 0 0 3s 9s 9s s 0 0 4s 9s 0 s 9s 9s 3s s 9s A:A, 0 Downloaded 03 Jan 003 to Redistribution subjet to AIP liense or opyright, see

5 96 J. Chem. Phys., Vol. 8, No., January 003 Fedin, Purtov, and Bagryanskaya 3 s 0 0 s 6 s 3s s 3 s 40 s 3s 3 s s 0 4 3s 6s 6s 0 3 s 3s 6 s 4 3s 0 3s 3s 0 s 6 s 6 7 s 3 s 0 s 3s R 6 7 s s 0 4 3s 3 s 40 s 3s 3 s 6s 6s, 0 6 s 4 3s s 3s 3s 3s 3s 6s 3s s 6s 3s 4s 9s 0 s 6s 3s 3s 6s 3s 0 4s 9s A:A 3 where s sin ( 0 ), s sin ( 0 ) sin ( 0 /). For a radial with two equivalent magneti nulei I / in a zero magneti field there are only 3 different eigenvalues of energy E,,3,4 a/, E 5,6 a and E 7,8 0 see 7, and thus transitions with only 3 different frequenies a/, a and (3/) a are possible see Fig. b. One an see from 3 that relaxational transitions between states 5, 6 and 7, 8 at a are not allowed in a zero magneti field. The relaxation transitions,, 3, 4 5, 6 ( (3/) a) and,, 3, 4 7, 8 ( a/) are allowed in the zero magneti field with probabilities given by the matrix Rˆ 0 in 3. The probabilities of all relaxation transitions are strongly dependent on magneti field, exept for transition 4 whih is forbidden in any magneti field. As an example, Fig. b shows the field dependene of relaxation rate onstants for transitions 6, 5 and 3 4, whih orrespond to 3 EPR transitions in the high magneti field. The probability of the transition 6, whih orresponds to the low-field EPR line e n n e n n at e a, dereases by a fator of 6 in the low magneti field e a. On the other hand, the transition 5 is forbidden in the high magneti field, but allowed in the zero magneti field. Figure b was alulated for a partiular realisti ase But, for any arbitrary 0 the magneti field dependene of eah relaxation transition probability is the same, with the only differene in its amplitude. Note, that for a radial with one HFI onstant in the zero magneti field, only relaxation transitions with 0 are allowed. But, for a radial with two equivalent HFI-onstants relaxation transitions at nonzero frequenies (3/) a and a/ are allowed in the zero magneti field. The partiular ase 0 0 refers to the situation where both nulei have the same oordinates. This never happens, but in this ase all above results on spin relaxation between eigenstates 6 (I ) apply to the relaxation in a radial ontaining one magneti nuleus with a spin I. Thus, for this radial the relaxation transitions in the zero magneti field are allowed at nonzero frequeny (3/) a, as opposed to the radial with one magneti nuleus I /. But, in general, the probabilities of relaxation transitions for the radial I in the low magneti field are markedly different in omparison with the high magneti field, and we expet this onlusion to apply to the radials with I as well. Note, that the strong magneti field dependene of the relaxation rate is not related to fators whih appear in high-field expressions for relaxation times e.g., and 0 and indeed depend on magneti field. Figures a and b were alulated for the limiting ase a, e, yet the field dependene is observed, whih is determined by the hanges in spin eigenstates. IV. SPIN RELAXATION CAUSED BY MODULATION OF ISOTROPIC HFI In this setion we onsider the spin relaxation aused by modulation of an isotropi HFI onstant. This mehanism of relaxation takes plae when the radial has internal motions aompanied by the hanges in its geometry, whih result in time-dependent alterations of isotropi HFI onstants. For example, this takes plae for ethyl, tertbutyl and other radials of similar struture. It is known that the HFI onstants on methyl hydrogens are slightly different, 7 therefore the rotation of the CH 3 -group around the C C bond results in flutuations of eah isotropi HFI onstant. Another mehanism of modulation of isotropi HFI in a tertbutyl radial was proposed to be the inversion of C-atom regarding the plane defined by the three methyl arbons 8 the pyramidal angle is equal to 7.4 9,0. Both mehanisms lead to rossrelaxational transitions ( m 0), whih have been studied previously in a number of papers 3 in magneti fields 50/ 300 mt. For a radial with one magneti nuleus the relaxation rate in a high magneti field an be alulated using the expression 4 a T r e, 4 where a(t) a(t) ā, ā is the mean value of the isotropi HFI onstant. The alulation of the isotropi HFI-indued spin relaxation in low and zero magneti fields has been done in the same way as in Se. III, following the steps desribed in Se. II. The spin-hamiltonian of stohastially modulated isotropi HFI in a basis 5 an be written as Downloaded 03 Jan 003 to Redistribution subjet to AIP liense or opyright, see

6 J. Chem. Phys., Vol. 8, No., January 003 Spin relaxation in low and zero field Ĥ t a t Ŝ Î a t 0 4C C 0 C C C C 0 4C C. The alulations show that in both high and low magneti fields only one relaxational transition 4 is allowed, with a rate: C C a T r e a. 6 It is obvious that for e a, 6 oinides with 4, but in the low magneti field e a the relaxation rate /T r 0, beause C C. Hene, taking orret aount of isotropi HFI-indued relaxation is neessary in a low magneti field. The magneti field dependene of the relaxation rate onstant is shown in Fig. 3 a for the ase ( e a ) ]. Sine /T r 0 in the zero magneti field, taking orret aount of relaxation is important in the alulations of CI- DNP and MARY, as for the anisotropi HFI-indued relaxation. It is well-known, that in the low magneti field a new EPR transition 4 appears for a rf-field parallel to the external magneti field. 5 For TREPR kinetis of this transition, taking orret aount of isotropi HFI-indued spin relaxation is ruial. Obviously, the kinetis should reflet the same trends as were alulated for 4 EPR transition in Ref. 9 Fig.. It is also expeted that taking orret aount of low-field isotropi HFI-indued spin relaxation is important for EPR transitions 4 and 3 4. For a radial with two equivalent magneti nulei with HFI onstant a: Ĥ t a t ŜÎ a t ŜÎ a t Ŝ z Î z a t Ŝ Î Ŝ Î a t Ŝ z Î z a t Ŝ Î Ŝ Î. 7 The alulations show that in both high and low magneti fields 6 relaxational transitions are allowed, with the following rates: 6 T C C C C a a r 8 9a 4 e 9a 4 e 6a e, 3 5 T C 3 C 3 C 4 C 4 a a r 8 9a 4 e 9a 4 e 6a e, 7 T r C C a a 8 5a 4a e 4 e a e 9a 4a e 4 e, T r C 3 C 4 a a 8 5a 4a e 4 e a e 9a 4a e 4 e, 6 7 T r C C a a 8 5a 4 e a e 9a 4a e 4 e, 5 8 T r C 3 C 4 a a 8 5a 4 e a e 9a 4a e 4 e. Downloaded 03 Jan 003 to Redistribution subjet to AIP liense or opyright, see

7 98 J. Chem. Phys., Vol. 8, No., January 003 Fedin, Purtov, and Bagryanskaya a ). At zero field the relaxation rate for both transitions is equal to zero /T r 0( e 0). This result oinides with that obtained above for a radial with only one magneti nuleus. Moreover, it is rather obvious, that these features should our in low and zero magneti field for a radial with any number of equivalent nulei, modulated synhronously i.e., when all a i (t) are equal funtions. Indeed, the stationary Hamiltonian of suh a radial in the zero magneti field an be written as Ĥ 0 a i ŜÎ i. 9 FIG. 3. Magneti field dependene of probabilities of relaxation transitions aused by the modulation of isotropi HFI for a radial with one magneti nuleus I / a and for a radial with two equivalent magneti nulei I /, a (t) a (t) b. Calulated urves orrespond to the probabilities of relaxation transitions between spin levels i and j as shown in the figure in the extreme motional narrowing limit. The probabilities of transitions 6 and 3 5 are proportional to ( a a ), while the probabilities of 4 other transitions are proportional to ( a a ). Therefore, the spin relaxation is strongly dependent on the harater of intraradial motions whih modulate the isotropi HFI. If both HFI onstants are hanged synhronously, i.e., a (t) a (t), only transitions 6 and 3 5 are allowed in both the high and low magneti field. If the modulation of both onstants is not synhronous, one need to alulate the exat values of averages ( a a ) and ( a a ). For example, if a a 0 os(wt) and a a 0 os(wt ), one obtains ( a a ) a 0 ( os ) and ( a a ) a 0 ( os ). Using the expressions 7 for C, C, C 3, C 4 we obtain that in a zero magneti field only two relaxational transitions 6 7 and 5 8 are allowed with a frequeny a. All the other transitions at frequenies a/ and (3/) a are forbidden. Note that this result is opposite to the one obtained for anisotropi HFI-indued spin relaxation in the previous setion. Therefore, the study of relaxation transitions in a zero magneti field an provide information on the ontributions of anisotropi and isotropi HFI-indued spin relaxation. Assume a (t) a (t) a(t), where only transitions 6 and 3 5 are allowed at any magneti field. Figure 3 b shows the magneti field dependene of relaxation transition probabilities for the ase ( e It is obvious that the eigenvetors of this Hamiltonian do not depend on a. Therefore, the modulation of a does not hange eigenfuntions. In other words, the perturbation is diagonal in the eigenbasis of the stationary spin- Hamiltonian. Thus, no relaxation transitions are indued. Therefore, the experimental study of isotropi HFI-indued spin relaxation in the zero magneti field an possibly give information on the harater of intraradial motions. For example, it is yet not lear whether the primary motion whih modulates isotropi HFI in tertbutyl radial is the inversion of a C-atom or the rotation of CH 3 -groups, 8, as mentioned above. In the former ase the modulation is synhronous, in the latter one it is not. The above onlusions for a radial with many equivalent nulei I / modulated synhronously apply also for radials with one magneti nuleus I /. In partiular, alulations for a radial with two equivalent nulei a (t) a (t); see 8 apply to the relaxation in radial with one magneti nuleus I for the eigenstates 6 whih orrespond to the total nulear spin I ). V. SPIN RELAXATION CAUSED BY MODULATION OF SPIN ROTATIONAL INTERACTION The literature data shows that for small alkyl and ayl radials e.g., CH 3 and HC O ] in nonvisous solutions modulation of spin rotational interation SRI is the primary relaxation mehanism 6,7 arising from the oupling of the eletron spin with the magneti moment of moleular rotation. In solutions the value of the total rotation momentum flutuates due to the frequent ollisions of the radial with neighboring moleules, and these flutuations lead to spin relaxation. For the ase of an axially symmetrial radial the relaxation time due to modulation of SRI in high magneti field e a an be evaluated as 8,9 I T T r C 3 C kt, 0 where C i are the omponents of the SRI tensor, I is the momentum of inertia, r is the radius of the radial, kt is the Boltzman energy, and is the visosity of solution. The omponents of the SRI tensor are unknown for the most of the radials, but they usually are estimated using the omponents of the g-tensor 30 and relaxation time is evaluated using the following expression 9 Downloaded 03 Jan 003 to Redistribution subjet to AIP liense or opyright, see

8 J. Chem. Phys., Vol. 8, No., January 003 Spin relaxation in low and zero field 99 T r g 3 g kt, where g i g i.003. For the onsideration of SRI-indued spin relaxation in a low magneti field we follow the same proedure as for anisotropi and isotropi HFI-indued relaxation. Both isotropi and anisotropi parts of SRI flutuate due to the ollisions of a radial with neighboring moleules, whose harateristi time ( s) is muh shorter than the time of radial rotation ( s). Therefore, modulations of both anisotropi and isotropi parts of SRI ontribute to spin relaxation in the same way, and we an neglet the anisotropi part in omparison with isotropi in alulations. Taking aount of the anisotropi part leads to insignifiant improvements of the result. For a radial with one magneti nuleus we shall operate again in a basis 5. The spin-hamiltonian of isotropi HFI in this basis an be written as ĵ z / C ĵ / 0 C ĵ / C Ĥ t C Ŝĵ t C ĵ / C C ĵ z / C ĵ / C C ĵ z 0 C ĵ / ĵ z / C ĵ / C ĵ / C C ĵ z C ĵ / C C ĵ z /, where C is the isotropi SRI onstant. The harateristi time of rotational momentum flutuations an be evaluated as: j I 8 r 3. 3 Averaging Ĥ (t) in we take into aount that ĵ z ĵ ĵ I kt. ĵ ĵ 4 Sine j s, the fators e j and a j in magneti fields mt for most of the radials and hene an be negleted. The matrix of relaxation R for the two limiting ases e 0(R 0 ) and e a (R ) in the Liouville basis,,,, 3,3, 4,4 an be written as / 0 / / 3/ / / R 0 0 / 3/ C / IkT j, / / / R 0 C 0 IkT j The probabilities of relaxational transitions at an arbitrary magneti field are P P 3 4 C C IkT j, P 4 P 3 C C IkT j, P 4 C C C IkT j. P 3 0, 6 One an see, that in a high magneti field only relaxational transitions are allowed ( 4 and 3 ), while in a low magneti field 5 different transitions are allowed ( 4, 3,, 3 4 and 4 ). The magneti field dependene of relaxation rate onstants is shown in Fig. 4 a. For spin levels and 3 the sum of the probabilities of transitions P P 3 P 4 P 3 P 3 P 3 4 C IkT j does not depend on magneti field. Therefore, the total relaxation rate between eah of these levels and all other levels does not hange, but is redistributed in the low magneti field. In zero magneti field spin states,, 3 are degenerate. One an see from 5 6 that the probability of relaxational transition,, 3 4 in a zero field is.5 times higher than of the high magneti field transition 4. This ours beause the new allowed transition appears between spin levels and 4 in the low magneti field. Its probability in the zero magneti field is equal to the probability of eah 4 and 3 4 transitions. Note that the transition 4 is forbidden in the high magneti field. Therefore, as for isotropi HFI-indued spin relaxation, taking the orret onsideration of SRI-indued relaxation in a low magneti field is neessary for a number of ases. In partiular, suh ases are i the alulation of TREPR kinetis in the perpendiular and parallel RF-field espeially for the latter, and ii the alulation of absolute values of CIDNP and MARY in mielles. For a radial with two equivalent magneti nulei similar alulations have been done for the same Ĥ (t) C Ŝĵ(t) in the basis of eigenfuntions 7. The probability of relaxational transition between spin levels 7 and 8 is P 7 8 C IkT j at any magneti field, all other transitions with partiipation of these levels are forbidden. The probabilities of transitions between other 6 spin levels are given by the following matrix in the Liouville basis,,,,, 6,6 : Downloaded 03 Jan 003 to Redistribution subjet to AIP liense or opyright, see

9 00 J. Chem. Phys., Vol. 8, No., January 003 Fedin, Purtov, and Bagryanskaya C C C C C C C 3 0 C C 4 C C 0 C C 3 C 3 C 4 C 4 C 3 C 4 C C 3 Rˆ 0 0 C 4 C 3 0 CIkT j. 7 0 C C 4 C 3 C 4 C 3 C 3 C 4 C C 4 C C C C C 3 0 C C 4 C C The analysis in zero and high magneti fields leads to similar onlusions as have been found for a radial with one magneti nuleus. New transitions between spin states 6 and 3 5 appear in a low magneti field Fig. 4 b, while the other relaxation probabilities are redistributed. Therefore, we onlude that for any arbitrary radial the rate of SRI-indued spin relaxation inreases in the low magneti field sine new transitions appear. The physial meaning of this effet is analogous to the influene of an osillating magneti field Ĥ (t) ŜBˆ (t). It is well-known that in the high magneti field B 0 EPR transitions are indued only if B B 0. But in a low onstant magneti field B 0, new transitions appear for B B 0. The appearane of new transitions indued by modulation of the SRI in a low magneti FIG. 4. Magneti field dependene of probabilities of relaxation transitions aused by modulation of the SRI for a radial with one magneti nuleus I / a and for a radial with two equivalent magneti nulei I / b. The alulated urves orrespond to the probabilities of relaxation transitions between spin levels i and j as shown in the figure in an extreme motional narrowing limit. field has the same ause and takes plae for any arbitrary radial. VI. CONCLUSIONS The above alulations of the spin relaxation of radials in solutions show that the probabilities of relaxational transitions are signifiantly different in low and high magneti fields. The use of high-field expressions with only the adjustment of e to the exat splitting between the energy terms is not orret. This applies for the relaxation mehanisms due to i modulation of anisotropi HFI, ii modulation of isotropi HFI, and iii modulation of spin rotational interation. These three mehanisms are dominant in a low magneti field, and taking aount of these orretly is important for many pratial ases. Anisotropi HFI-indued spin relaxation has been onsidered i for a radial with one magneti nuleus I /, 9 ii for a radial with one large HFI onstant and several small additional onstants, 9 iii for a radial with two equivalent HFI-onstants, and for a radial with one magneti nuleus I. For all studied ases it is shown that the probabilities of relaxational transitions are markedly different in high and low magneti fields. The most pronouned differene in relaxation times is observed for a radial with one HFI onstant (I /) or with one large and several smaller HFI onstants i ii. The less distint differene is observed for a radial with two equivalent nulei I / or with one nuleus I. For a radial with numerous HFI onstants the relaxation is very omplex even in a high magneti field, therefore one would expet that the differene between relaxation in a high and low magneti field beomes not so obvious. Consequently, we ome to the general onlusion, that anisotropi HFI-indued spin relaxation shows the speifi features for radials with a small number of magneti nulei or in the ase when several HFI onstants signifiantly exeed all other onstants ; the more HFI onstants a radial ontains, the less pronouned are these features. Nevertheless, it should be noted that for any arbitrary radial the probabilities of the relaxational transitions are different in the low and high magneti field, and simple use of high-field expressions for e 0 is not orret. Isotropi HFI-indued spin relaxation has been onsidered for a radial with one HFI-onstant and with two equivalent HFI onstants. For a radial with one magneti nuleus the rate of spin relaxation dereases in a low magneti field and is equal to zero in a zero magneti field. The same feature ours for a radial with two equivalent HFI Downloaded 03 Jan 003 to Redistribution subjet to AIP liense or opyright, see

10 J. Chem. Phys., Vol. 8, No., January 003 Spin relaxation in low and zero field 0 onstants whih are modulated synhronously, and this is also valid for a radial with any number of equivalent nulei modulated synhronously and, finally, for a radial with one nuleus I /. In synhronous, as well as in all other haraters of modulation, the rate onstants of spin relaxation transitions strongly depend on magneti field, whih should always be taken into aount. SRI-indued spin relaxation has been onsidered for a radial with one HFI-onstant and with two equivalent HFI onstants. In these ases and for any arbitrary radial the rate of SRI-indued spin relaxation inreases in low magneti field due to the appearane of new allowed transitions. In onlusion, we would like to note that the radials with one large HFI onstant and several smaller onstants are found rather often, espeially in reations of isotopi abundant ompounds. The radials with a set of equivalent nulei are even more ommon. Therefore, both quantitative and qualitative results of this work are important for many experimental studies in a low magneti field. ACKNOWLEDGMENTS The authors thank Dr. A. G. Maryasov for fruitful disussions. This work was supported by the Russian Foundation for Basi Researh Grant No. N a, by IN- TAS , and by the Siene Support Foundation, grant for talented young researher. L. T. Muus, P. W. Atkins, K. A. MLauhlan, and J. B. Pedersen, in Chemially Indued Magneti Polarization Reidel, Dordreht, 977. K. M. Salikhov, Yu. N. Molin, R. Z. Sagdeev, and A. L. Buhahenko, Spin Polarization and Magneti Effets in Radial Reations Elsevier, Amsterdam, U. E. Steiner and T. Ulrih, Chem. Rev. 89, B. M. Tadjikov, D. V. Stass, and Yu. N. Molin, J. Phys. Chem. A 0, N. C. Verma and R. W. Fessenden, J. Chem. Phys. 65, J. S. Jorgensen, J. B. Pedersen, and A. I. Shushin, Chem. Phys., E. G. Bagryanskaya and R. Z. Sagdeev, Prog. Reat. Kinet. 8, C. R. Timmel, U. Till, B. Broklehurst, K. A. MLauhlan, and P. J. Hore, Mol. Phys. 95, M. V. Fedin, P. A. Purtov, and E. G. Bagryanskaya, Chem. Phys. Lett. 339, A. G. Redfield, in Advanes in Magneti Resonane, edited by J. S. Waugh Aademi, New York, 965, Vol., pp. 33. G. Breit and I. I. Rabi, Phys. Rev. 38, N. Bloembergen, E. M. Purell, and R. V. Pound, Phys. Rev. 73, I. Solomon, Phys. Rev. 99, R. Freeman, S. Wittekoek, and R. R. Ernst, J. Chem. Phys. 5, E. G. Bagryanskaya, H. Yashiro, M. V. Fedin, P. A. Purtov, and M. D. E. Forbes, J. Phys. Chem. A 06, M. V. Fedin, H. Yashiro, P. A. Purtov, E. G. Bagryanskaya, and M. D. E. Forbes, Mol. Phys. 00, H. Fisher and K. H. Hellwege, in Landolt Bornstein, New Series, Group Springer-Verlag, Berlin, 977, Vol.9. 8 P. W. Perival, J.-C. Brodovith, S.-K. Leung, D. Yu, R. F. Kiefl, G. M. Luke, K. Venkateswaran, and S. F. J. Cox, Chem. Phys. 7, M. Yoshimine and J. Paansky, J. Chem. Phys. 74, M. N. Raddon-Row and K. N. Houk, J. Am. Chem. So. 03, E. G. Bagryanskaya, G. S. Ananhenko, T. Nagashima, K. Maeda, S. Milikisyants, and H. Paul, J. Phys. Chem. A 03, G. H. Goudsmit, F. Jent, and H. Paul, Z. Phys. Chem. Munih 80, P. P. Borbat, A. D. Milov, and Yu. N. Molin, Chem. Phys. Lett. 64, H. Kurrek, B. Kirste, and W. Lubitz, Eletron Nulear Double Resonane Spetrosopy of Radials in Solution VCH, Berlin, A. Carrington and A. D. MLahlan, Introdution to Magneti Resonane with Appliations to Chemistry and Chemial Physis Harper & Row, New York, D. M. Bartels, R. G. Lawler, and A. D. Trifuna, J. Chem. Phys. 83, H. Paul, Chem. Phys. Lett. 3, P. S. Hubbard, Phys. Rev. 3, P. W. Atkins and D. Kivelson, J. Chem. Phys. 44, ; P.W. Atkins, Mol. Phys., R. F. Curl, Mol. Phys. 37, ; 9, Downloaded 03 Jan 003 to Redistribution subjet to AIP liense or opyright, see