Theory of geminate recombination of radical pairs with instantaneously changing spin Hamiltonian. III. Radical recombination in switched high magnetic field

Chemical Physics North-Holland

166 ( 1992) 35-49

Theory of geminate recombination of radical pairs with instantaneously changing spin Hamiltonian. III. Radical recombination in switched high magnetic field S.A. Mikhailov Institute for Water and Environmental Problems, Slberlan Branch of the Russran Academy of &ewes,

P.A. Purtov

Barnaul656099,

Russia

and A.B. Doktorov

Instrtute of Chemrcal Kinetrcs and Combustion, Siberran Branch of the Russran Academy of Sciences, Novosrbwsk 630090, Russia Received

I8 February

1992

The theory of the recombination of radical pairs (RPs) in a switched hrgh magnetic field is proposed for the first time. The RP recombination probability is calculated for a stochastic motion of reagents in solution. Within the framework ofa phenomenological two-positional model the analysis of the external field switching effect upon the recombination probability and CIDNP effects in the reaction products is carried out. It is shown that the switched magnettc field experiments contam mformation on the RP magnetic-resonance parameters as well as on kinetic ones. The above experiments are compared with the time-resolved ones, and the “effectiveness” of this kind of experiments is discussed.

1. Introduction The formal mathematical method described in ref. [ 1 ] is applied to calculate the recombination probability and nuclear spin polarization in the products of radical reactions proceeding in a switched magnetic field, depending on time as follows: B(t)=B, =Bz,

)

oto.

(1)

Here switching time, to, is less than the time typically for geminate processes to take place. This kind of experiments is known [ 2,3], but a theoretical description of the effects observed is not available in the literature on the subject. In the present paper the theory of geminate recombination of a radical pair (RP) in a switched magnetic field is developed for the first time. In section 1 the problem is formulated on the basis of the results obtained earlier [ 11. In sections 2 and 3 we calculate the recombination probability of radical pairs in a switched high magnetic field (eq. ( 1) ). The formulae obtained can be adopted to calculate the effects under discussion in the RP with different types of relative motion of particles. Section 4 presents concrete calculations within the framework of the phenomenological twopositional model. In section 5 we review the problem concerning the information we can get from experiments with switched fields. Finally, in section 6 we compare two types of experiments in terms of the information they provide: experiments with a switched field and a time-resolution one. According to a generally accepted approach to describe the spin state of a radical pair let us introduce a density Correspondence to: P.A. Purtov, Institute Novostbirsk 630090, Russia. 0301-0104/92/$05.00

of Chemical

Kinetics and Combustion,

0 1992 Elsevier Science Publishers

B.V. All rights reserved.

Siberian

Branch of the Russian Academy

of Sctences,

S.A. Mikhador

36

et al. /Chemical

Physics 166 (1992) 35-49

matrix, p( t, to), on the basis of all electron-nuclear wavefunctions of the RP. Generally speaking, this matrix depends on both the current moment, t, and the moment, to, when the external magnetic field is switched. The state of a spin subsystem of the product molecules formed as a result of RP recombination is defined by the matrix (p(t', J 0

w(t, to)=Uo

to)) dt’ .

(2)

The diagonal components wNNof this matrix define the probability of RP recombination from the spin state IN). It should be noted that expression (2) involves (p( t, to) ) - the density matrix averaged over spatial coordinates within a reaction zone (a certain volume favorable for radical recombination). U. is a superoperator; its matrix elements characterize the “recombination rate” of different components of the RP spin density matrix (see ref. [ 1 ] for details). Formally, the switching of the external magnetic field means “instant” change of recombining the RP spin Hamiltonian. So let us make use of the results obtained in ref. [ 11. According to formulae (62) of ref. [ 11, the double Laplace transform of w( t, to) over both variables is determined as follows: ~(s,u)=s-‘Uo{u-‘G,(s+u)+[1+G,(s)W,]-’G,(s,u)}[1+W,G,(s+u)]-’po. Here p. is the density matrix characterizing Wo=Uo+iVo

(3)

the initial state of the RP. The operator (4)

describes (in the contact approximation) the recombination of radicals and their exchange interaction. The operators G,(p) (k= 1, 2) and G3(.s, u) are determined by the spin dynamics in the RP and by the relative motion of particles:

dt , k=l,2,

(5)

jK(t)exp(--St)dtjT,(t-r)T,(r)exp(-ur)dr.

(6)

G&J) = 5 K(t)T,(t)

exp( -pt)

0 m

G3(.s,u)=

0

I

0

In the above expressions Tk( t) = exp( i&t) is the operator of the RP spin evolution, and & is the Liouvillian describing the spin interactions in RP before (k= 1) and after (k= 2) time to of magnetic field switching. The function K(t) characterizes the relative motion of the radicals of the pair [ 11. The ranks of the matrices in eq. ( 3 ) are defined by both the number of the spin states of the electron-nuclear system (RP) and its symmetry, and if the RP is composed of radicals with many magnetic nuclei the ranks appear to be very large. However, eq. (3 ) can be solved analytically for an arbitrary number of magnetic nuclei provided that the S-To approximation is used [ 41. Within the framework of this approximation the RP spin Hamiltonian includes the Zeeman energy of unpaired electrons and the secular part of the isotropic hti, (7) =gACB#B, B is the external magnetic field, s,.,A(Bjand fkk(n)are operators of the electron spin and Here ~~~~~~~ that of the kth (nth) nucleus which belongs to radical A(B), aAk(B,,)is the corresponding hfi constant (radians per second). The singlet-triplet (S-T,) transitions within the RP induced by the spin Hamiltonian (7) do not change orientation of nuclear spins. Therefore, the RP ensemble may be divided into subensembles with fixed nuclear

S.A. Mikhailov et al. /Chemlcal Physrcs 166 (1992) 35-49

spin configurations {m} = {m,, m2, .... m,} (N is the total number of magnetic may calculate the formation of the product molecule with a certain configuration stages of this probability calculation are described in the next two sections.

37

nuclei in the RP), and then we of nuclear spins {m}. The main

2. The main formulae Let us consider the RP subensemble with a certain nuclear spin configuration {m}. It has already been mentioned that a singlet-triplet evolution in each subensemble is caused exclusively by the S-To transitions, while the initial populations of the triplet sublevels T_ and T, remain unchanged. So, in order to solve the problem it is sufficient to consider only the part of the RP density matrix where changes take place. Let us choose the functions IToTo),

IToS),

ISTo),

(8)

I=>

as a basis for further consideration, and write down the explicit form of all superoperators involved in eq. (3). Firstly, consider the operator U. determining the rate of recombination of radicals being in contact. Usually, when the spin effects in radical reactions are estimated, the assumption is made that the RP recombination is possible only from a singlet electron state (the so-called “spin selective rule”) [ 41. In the present paper, however, we shall not limit ourselves by defining a concrete reactable multiplet RP state beforehand, but introduce the rates of recombination from both states, Ks and KT. This allows us, firstly, to apply the theory to investigate the influence of the correlation of the different recombination channel rates to the spin effects, and, secondly, to analyze the spin effects in systems where product molecules are formed mostly from the triplet RP [ 5 1, and thirdly, to use the results obtained for reactions in which there is no spin prohibition (Ks = KT), but the product spin state manifests itself, for instance, by the luminescence of certain multiplets [ 2,3,6]. On the basis of functions (8) the operator matrix U. is diagonal and its elements are KT u = 0 0 0 i: 0

0

0 0

(&+&)/2 0 0

(K,+Ks)/2 0

0 0 0 ’ KS I

(9)

The exchange interaction of unpaired electrons plays an important part in the RP spin evolution. Just this interaction is responsible for the splitting of the RP electron levels into singlet and triplet states. In the theory that has been developed in ref. [ 1 ] the exchange interaction, as well as recombination, is considered in the contact approximation, i.e. the exchange integral Jo is assumed to be non-zero exclusively within the reaction zone. The operator Vo, which describes this interaction, on the basis of functions (8) is a diagonal matrix, /o

0

0

0

0

00

o\

Obviously, the operator W defined as the sum (eq. (4) ) of operators U. and V. is also diagonal. The superoperator Tk( t) describing the RP spin dynamics in magnetic fields Bk (k= 1, 2) is the following matrix on the basis of functions (8):

c!?(t) Tk(t) = i

ick(t)sk(t)

-ick(fbk(t)

-Ck(tbk(t)

c:(t)

si(t>

-iCk(tbk(t)

s:(t)

c:(t)

sift)

-iCk(tbk(t)

iCk(fbk(t)

(11)

S.A. Mikharlov et al. /Chemical Physics 166 (1992) 35-49

38

Here c~(~)=cos(o~~/~),

k=l,

Sk(t)=sin(mkt/2),

Ri=hk-&3)/-@-‘&+

c %)m,t

2,

c 63nm. n

(12)

wk is the frequency of S-T,, transitions in the subensemble under study. It depends on both the nuclear spin

configuration {m> and the external magnetic field amplitude. Substituting eq. ( 11) into eq. (5), we find matrices of operators G,(p) and G*(P):

G(P)

=

&)/2+&!(P) Ck (PI -G(P) i tP)n-tk+(P)

,,,,:5ZJ;

+

(p)

g~P~,2-~~~P~ -5k (PI

where Ck’(p) = [g(p-iwk) kg(p+ the motion of the radicals [ 11,

g(p-F_(;)(p)

&)/2-C(P)

g~P~,2+~~~P~ 5k (PI

-t(k fP) 5k(P) ’ g(P)/2+<,+ (PI 1

+

(13)

iok) ] /4, while g(s) is the Laplace transform of K( t), which characterizes

00

K(t)exp(-st)dt.

114)

0

To calculate the elements of the operator G3(s, u) matrix we use formula (6). But instead of direct multiplication of matrices let us act as follows [ ‘71. Determine the eigenvalues ke, and Ae2of the operator (7) and a transformation matrix from functions of singlet-triplet basis IZ) ( IZ) = ITO), IS ) ) to the basis of eigenfunctions (A) of this spin Hamiltonian. (Further we denote the Hamiltonian eigenfunctions by Greek letters, and the functions of the S-To basis by capital Latin ones.) The matrix elements z+,= (ZIJ.) of the above transformation matrix do not depend on the external magnetic field amplitude (hence, they are not changed at the field switching). This allows us to express both operators T, (t) and Tz (t) via the elements vM, (T,Jt) )K,MN= I: v,&~v&~uv,, exp( - it,$) t ) , AP

(15)

where e,$,“)=eJk) - E$~)is the splitting of the eigenlevels of the RP subensemble with a fixed nuclear spin configuration {m} in the magnetic field Bk (k= 1, 2). Note, that all matrix elements vip are real, thus, the complex conjugation sign “*” is omitted below. According to formula (6) to find the matrix elements of the operator G3(s, u) it is necessary to calculate the following value:

Here the matrix elements of the product of three superoperators are presented as a sum of products of their matrix elements. Let us calculate the integral over the variable T in this expression. In conformity with eq. ( 15) & (T,( -5) )PQ,KL(TI(r) )KWV

(17)

The summation of K, L can be done in the above expression. Using the unitarity of the real matrix {v~},

S.A. Mikhailov et al. /Chemical Physrcs 166 (1992) 35-49

39

(18)

and taking the summation 2 (T,(-r)

over v, q we obtain

)PP,KL(TI(~))KL,MN=

Consequently,

C b

(19)

v~~v,~v~v,,exp(itft'~-i~f:'t).

for the integral to be found we have

I

s

exP(-ur)dr

PQ.KL(TI(T))KL,MN

LVz(-W

0

=Ev,,V~QV1MV~(~+iE~~‘-iEj~‘)-‘[

1 -exp(

.

-_ut-iitJi’t+ieJ:‘t)]

(20)

Now, let us substitute Expression (20) into formula ( 16)) where the operator T2 (t) matrix elements are also presented as the expansion ( 15 ). Using matrix unitarity once again and performing the summation analogous to that giving formula ( 19 ) , we find

02

x

s 0

=E

K(t) [exp( -st-iej:)t)-exp(

--ut--st-iej:‘t)]

dt

.

vUv~vLMvpN(~+i~J~)-itJ~))-1[g(~+it~~))-g(~+~+i~J~))]

Formula (2 1) involves only four terms (since A, p= 1, 2 ) and is very convenient ator G3 ( S, u) matrix elements. Using eq. (2 1) we find that +/2+d+ d-

G,(s, u> =

-d$/2-d+

d@/2+d+ @/2-d+ -d-

where 9 = @(s, u ) is a function @(s,

d’=

[@(s-iw2,

’

of the oper-

(22)

@/2+d+

9

d * is expressed in terms of this function u+i@It@(s+iw,,

while 6 is the S-To transition 6=w* -0,

+/2-d+ -dd-

for calculations

of two complex variables,

~)=~-‘k(~)-gcs+~)l

and the parameter

-d@/2-d+ @/2+d+ d-

(21)

u-iS)]/4,

frequency

change which occurs at the magnetic field switching

.

(23)

3. Calculation of the RP recombination probability in a switched magnetic field Now we can proceed to direct calculations

of the elements of the matrix fi ( S, u) by formula

( 3 ). Inverting

of

40

S.A. Mlkhailov et al. Khemcal

Physrcs 166 (1992) 35-49

the matrices is a certain problem here, however, it is simplified by the following circumstances. Firstly, it is not necessary to calculate all elements of the matrix &(s, U) in order to find the recombination probability of a given RP subensemble; it is sufficient to calculate either &,T,, and &s depending on what RP multiplet state is reactant. Secondly, in the density matrix pO,which describes the RP subensemble spin state at the initial time, all non-diagonal components are equal to zero, while diagonal elements are non-zero. This means that in the basis of functions (8) p. can be represented as a column vector: Po=(%rO,O,%).

(24)

Note, that if the RP is formed as a results of dissociation of a triplet molecule, then only the first element of the vector (24) is non-zero. And vice versa, if the RP-precursor molecule was in a singlet state, then the only nonzero element of the vector (24) is ns. It follows from the above that for the construction of the inverse matrices in expression (3) it would be sufficient to calculate the elements of the first and fourth rows of the matrix [I+ G, (s ) W, ]- 'and the elements of the first and fourth columns of the matrix [1 + VVOG, (s+ u ) ] - ‘. Actually the amount of calculations required even half as much, because these matrices are mutually transposed (neglecting denotations), as is easily concluded (see eqs. (9 ), ( 10) and ( 13 ) ). The probability of the RP subensemble recombination from a singlet state is characterized by the element Ms. u), PZ?(fo, t)+%,ss(& u) .

(25)

(By the sign + we denote the correspondence between the original function and its double Laplace transform over both variables. ) Being in a triplet state, RPs form product molecules with the probability PTm(to,t)+&0To(% u) 1

(26)

in case their recombination is not prohibited. However, the component GToTO characterizes the recombination of pairs from the state To only, and does not take into account those from the two other triplet states of the RP, T-i and T,,, which are not involved in the RP singlet-triplet evolution by the Hamiltonian (7). If at the moment of RP formation these levels were populated, n-,-*# 0, they would contribute to a recombination probability of the triplet RP too. This cont~bution can be easily taken into account. The spin functions T_ , and T, , are eigenfunctions for the spin Hamiltonian (7). Thus, we have a scalar expression (not a matrix one) to calculate the matrix elements &_ ,T_, and &, ,r+, by formula ( 3 ) . Carrying out the calculations required we found that the following formula should be used (instead of eq. (26) ) to calculate the recombination probability of the triplet RP: PT,(fO,

+‘GToT~(h

u)+

(su)-‘(nT+,

+nT.-,

)KT&s)/

[ 1 +KTg(s)

1 .

(27)

For the elements of the matrix &(,, u) which characterize the RP recombination probabilities the following expressions are obtained:

#T)~oTo(h

U)=KT(CT~SY~S+CT~T~~T~)

,

(29)

where (30) (31) (32) (33)

S.A. Mkhallor et al. / Chenucal Physm 166 (1992) 35-49

41

dk=(ek’+Ks)(eK’+KT-V)k)+(ek’+KT)(ek’+Ks-~k))

(34)

~k=t[(Bk’-uk1)+(ek1-6k’)])

(35)

D= (al’

-aT’)/(u+iS)-

0, =g(s+u) h =a)

, ,

(b,’

--b2’)/(U-i@

a, =g(s+u-iw,) u2 =g(s-iw2)

, ,

,

(36)

b, =g(s+u+iw,) 6, =g(s+&)

, .

(37)

while the function g(p) is defined by expression ( 14). The first terms within curly brackets in eqs. (30)-( 33) do not depend on the parameters u and 6. They characterize the subensemble recombination probability in magnetic field &. The switching effect to be found is completely defined by the second component, which reduces to zero in the case of a steady field (B, = B2), because 6 is equal to zero. Only formulae corresponding to the situation of a strong exchange interaction which is turned on when the (J o-+co) is of the utmost practical radicals get into the reaction zone are given here. Such an approximation interest and it essentially simplifies expressions for recombination probability. We use this approximation, because former calculations for a static magnetic field have shown that it describes the spin effect quite well [ 4,8]. For further simplification let us consider the following special case which is often realized in experiments. Let the RP be formed as a result of the triplet molecule dissociation, i.e. in the initial density matrix (24) n,=O,

n,,=1/3Z.

where Z= n,N=, (21, + 1) is the total triplet RP is prohibited (K= = 0 ) and that the reaction rate of the radicals recombine without failure (&-co). solely by the component (28 ) which ~~s(s.u)=(3Zsu)-‘[i-2/4;+iSe,o/(d;d;)],

(38) number of nuclear spin states of the RP. Suppose the recombination of a the reaction product is formed by singlet RPs only. Besides, let us assume is so high that all RPs which get into the reaction zone in a singlet state Then the probability of the RP subensemble recombination is defined is equal to d;=i+f3k(akl+b~1)/2,

(39)

under the assumptions made. Despite the simplifying assumptions, formula (39) is still too complicated; thus, it is not easy to draw any conclusions about the effect of the external magnetic field switching upon the yield of radical reaction products. In order to illustrate this effect we consider a particular example.

4. A two-positional model A type of the molecular motion of radicals is not defined exactly in formula (39). This motion is considered to be described by a certain time function K(t) or by its Laplace transform g(s). Note that for some models of the particles’ molecular motion the functions g(s) have already been calculated in the literature (see ref. [ 1 ] and references therein). By choosing a model of radical motion in solution and, substituting the corresponding function g(s) to formulae ( 37 ) and (39 ), one can obtain the expression for the RP recombination probability accounting for a specific type of the relative motion of the particles. To describe the molecular motion of the reagents we use in the present paper (as well as in ref. [ 1 ] ) a phenomenological RP model known as the twopositional model [ 9, lo]. This model considers the process of the approach of the radicals up to the distance of closest contact (where radicals can recombine), the radical escapes from the contact and the re-encounters as the conversion of the certain quasi-particle, namely, the RP. The first RP state corresponds to that of closest contacting radicals. This state is characterized by the rate 1/r, of the RP turning to the second state, conforming to the spatially separated radical state. From the latter state the RP can come back (to the first state - re-

42

S.A. Mikhailov et al. /Chemical Physm 166 (I 992) 35-49

encounters) or dissociate. These processes are described by the rates 1/rZ and 1/rC, correspondingly. tion g(s), which conforms to this model of radical motion, has been obtained in ref. [ 11: s+l/r g(s) =

(40)

’

S(S+l/t)+(S+l/TC)/T,

The func-

where l/r=l/r,+l/r*.

(41)

Using this function (40) let us investigate the external field switching effect on the total yield (not on the RP recombination kinetics) of the reaction product. With this purpose the quantity FM&) = lim &(t,

to)

has to be calculated for every RP subensemble. limiting transformation directly in expression p,(&)+

lim [s&(s,

limiting theorem allows us to execute the

.

u)]=Css(u)

s-0

The Laplace transform (39):

(42)

By substituting (40) into the expression obtained that within the framework of the model chosen

for tiss(s, u) and making the transformation

required we find

~~s~u~=~o,[u-‘-~,~u~l, F,(u) =

(1-o,l~*)(U+llt)~U+(1+W1/~2)/~1 u{&(n-

1)/2+

[ 1 +uz,+u7,(

1 +Nri)]

[(u+

l/r)2+0:]}

(43)

’

where

(w2t)“(nwom= 3Z[l+(oar)‘(n+1)/2]

I)/2

w,, does not depend on the parameter u, and it is the RP subensemble netic field B2. In eqs. (43) and (44) we used the dimensionless parameter n=r,/r=

(44)

’

1 +rC/r2.

recombination

probability

in the mag-

(45)

The physical meaning of this parameter is obvious. By definition, 7c is the RP lifetime, while ~~ is the average time of RP conversion into a closest contact state. Thus, the ratio r,/r, can be naturally called the average number of re-encounters, and n_( eq. (45 ) ) the total number of partner contacts if the recombination process is “turned off”, i.e. if RP destruction is determined by the process 1/rC only. Another newly introduced parameter in eq. (43) is rr. It characterizes the total radical encounter time (i.e. the time of the radicals being in the reaction zone), and it is the product of one encounter time, rl, and the total number of radical encounters, n: rr= ri IZ. While calculating the function p, ( to) (eq. (42) ) by using eq. (43 ), let us pay attention to the processes which are proceeding during a time interval longer than that of one encounter: to > TV.Then in eq. (43) we can neglect the term UT, (because within the latter approximation u7] < 1 ), and as a result we obtain in the denominator of eq. (43) instead of the expression within brackets the third degree polynomial R3(U)=[U+1/(?~++~)][(U+1/7)2+0:]+to:(n-1)/(r,+?,).

(46)

To find the roots of R3 (u) let us substitute u=-

l/r+C+z,

(47)

S.A. Mikhailov

et al. /Chemxal

1/ (7, + 7r) ] ) into the cubic equation

(where l= f [ l/7form

43

Physics 166 (1992) 35-49

R3 ( u) = 0. It allows us to reduce the latter to the short

z3+3pz+2q=o, p=o:/3-{2) 4=

-t[(W1/2)~+<‘1-

(w,/2)2(7,/7)/(7,+7,)

.

(48)

The analysis shows that eq. (48) has solely a real root, which is equal to (Cardano

solution)

z, =A+B,

(49) B= ( -q-Jm)1/3.

A= ( -q+Jp+42)“3, Consequently,

R3 ( U) can be represented

in the form

R,(u)=(u+b)[(~+a)~+o~], where a=b+3z,/2,

b=l/r-r-z,

By substituting find

the latter expression

Pm(h)=%m[l-fm(~o)l

fm(h3)= Ll,“;/P; c

o’~3(A-B)~/4.

,

(50)

for R3( U) into eq. (43) and using the Laplace transform

tables [ 111 we

7

sin(Oto)exp(-at,)+P,{1-exp(-a?,)[cos(oto)+(a/w)sin(wt,)]} r

-Qmexp(-bto)(l-exp[-(a-b)t,]{cos(ot,)+[(a-b)/w]sin(oto)}). The function fm(to) is the reverse Laplace transform used in eq. ( 5 1) for brevity:

Pm=

1-0:/w: l+(o,r)‘(n+l)/2’

e m

(51) of the function

= (l-o,/02)(l+W,/02-b7)(l_bt) b.r2(7c+7r)[(a-b)2+w2]

F,(U).

’

The following designations

are

(52)

Formula (5 1) defines a RP subensemble recombination probability which depends parametrically on the time t,, of external magnetic field switching. The total recombination probability of radical pairs and CIDNP effects can be expressed in terms of eq. (5 1) by the known formulae (e.g., see ref. [ 41). Thus, we can conclude that the question of the external magnetic field switching effect on the RP recombination probability has been answered within the framework of the phenomenological two-position model. And now let us discuss what additional information can be obtained from the experiments on radical reactions proceeding in the switched magnetic field in comparison with those carried out in the static field.

5. Extraction of information from switched field experiments To begin with, consider a RP having no magnetic nuclei. This RP spin evolution occurs exclusively through g value differences of the unpaired electrons of the radicals. Therefore, ok= (gA-gB)/3Bk/fi (k= 1, 2). There is a sole RP subensemble here, and formula (5 1) defines the total recombination probability of radical pairs (for this reason we omit the subscript m below). At to = 0 we obtain the RP recombination probability in magnetic field B2, while at fO+cc we obtain that in the field B,. The whole dependence on the switching moment to is

S.A. Mkhadov et al. /Chemical Physrcs 166 (1992) 35-49

44

determined by the functionf( to). Hence, this function can be defined provided that p( to) and w. are measured experimentally. Thef( to) behaviour is determined by two “relaxation” times ( 1/a and 1lb) and oscillation frequency w. All these parameters are defined by the parameters of radical molecular motion (within the framework of the considered model, by times r,, r2 and r,) and by the S-To conversion frequency in the field B, before switching (see eq. (48)-( 50) ). Iff( to) is measured and the parameters u, b and o are calculated, then certain conclusions can be made concerning the character of the relative motion of the radicals (specifically for the present model, it allows to determine the time interval between re-encounters. TV, or the RP dissociation time, r,) and the frequency of S-To transitions can be calculated. Let us consider some ultimate cases in detail. If the mean number of radical re-encounters is fairly small, n - 1 zz 1 (i.e. r, % T2), two different recombination regimes are possible: (i) w1r2 B 1 and (ii) ~~‘5~< 1. The first one corresponds to fast RP singlet-triplet evolution (compared to relative motion), while the second regime corresponds to the opposite case. Supposing the reasonable condition 75,c 7, is met, the following expressions are obtained for case (i): a-r,’

+‘r-’ 42

>

bxz,’

+jT;’

,

ozo,

.

(53)

As one would expect, the “relaxation” times, 1/a and 1lb, do not depend on the RP magnetic resonance parameter o, and are determined by the kinetic parameters only. At the same time, the oscillation frequency does not depend on the kinetic parameters, and coincides with the S-To conversion frequency in the field B1. Within regime (i) quantum oscillations should be explicitly pronounced. For regime (ii) the following results are obtained: uzr,’

+r,’

=r-’

,

bz=r,’

,

coxo,/~.

(54)

As well as regime (i), the kinetic and magnetic resonance parameters of RP are independent, but within the present regime quantum oscillations cannot be observed, since the “relaxation” time T= 1/a is not long enough for appreciable singlet-triplet evolution. Now let us turn to the case IZ>> 1, which is more interesting from a practical viewpoint. Three different recombination regimes are possible in this case: (i) o,r2 > 1, (ii) m1r2 < 1 and w,r,/&z > 1 and (iii) o,r,/& << 1. Within regime (i) the RP S-To evolution is fulfilled during the time interval between two re-encounters, and, consequently, the “relaxation” time observed becomes independent of the RP dissociation time r,. For this case calculations give the following results:

f(to)~(w,rC)-‘(l-~w,/02)exp(-uto)sin(o,to).

(55)

The functionf( to) is the dying oscillation with frequency w, and characteristic damping time 4~~/3. Iff( to) is measured by experiment, then it is quite easy to determine both the frequency S-T, transitions and the time interval between re-encounters. If the initial amplitude of the oscillations is being measured too, one can calculate the RP dissociation time and find the number of re-encounters. Within the second regime the time interval between re-encounters is too short for the singlet-triplet transitions to be fulfilled, but they are effective during a pair’s lifetime (RP dissociation time). Note, that S-To mixing effectiveness is determined by the parameter w,rJ&z (but not by w,T,); this is connected with the fact that RP spin evolution is interrupted many times during the lifetime of the RP by partner re-encounters. For case (ii) we obtain uzt;’

,

bz:o:T2/2,

f(to) z2(wftct2)-‘(

o~~,/fi, 1 -w:/w:)

As one would expect, the shorter

[ 1 -exp(

-bt,)]

“relaxation”

. time

(56) l/u

is independent

of the singlet-triplet

transition

fre-

45

S.A. Mlkhadov et al. / Chemrcal Physics 166 (1992) 35-49

quency, but it is not manifested in the intermediate process function f( to). Quantum oscillations are also not pronounced here, because w,r2 CK 1. However, the second “relaxation” time, 1lb, depends on the S-To conversion frequency. Hence, as follows from eq. (55) this time can be determined by experimental measurement f( to). But b depends on both o, and 72, and consequently it is impossible to find simultaneously both the S-To conversion frequency and the time interval between re-encounters. Finally, consider the recombination (iii) regime which corresponds to the condition o,/~,/&-=K 1. Here magnetic resonance and kinetic parameters are fully “uncoupled”: bz7;‘,

az7,‘,

cmci_~,/Ji,

f(tO)=(l-0$/wZ)[l-exp(-t,/r,)].

(57)

An interesting regularity of the present regime is that it allows us to determine experimentally the RP lifetime, 7,. Under the above conditions the S-T,, transitions have no time to occur, therefore quantum oscillations cannot be observed. It is of interest to note that one can easily determine by experiment which regime of the geminate recombination process proceeds. Pronounced quantum oscillations mean that we are concerned with regime (i). Iff( to) is an exponentially decreasing function, but the damping decrement changes at external magnetic field alternations, we can state the regime as the second one, i.e. (ii). And regime (iii) is characterized by the functionf( to), damping decrement independence of the external magnetic field. Note that in experiments one can control (to a certain extent) the recombination regime, for instance, by changing the viscosity of the solution. If the viscosity of the solution raises, the effectiveness of spin transitions increases, and as a result, the third regime can transform into the second one, and the latter into the first one. Above we have considered a RP having no magnetic nuclei. However, radical pairs contain usually several magnetic nuclei, thus, the total RP recombination probability has to be represented as the sum of recombination probabilities of different subensembles. This case being too complex, a detailed analysis is possible for concrete RP only, if one disposes preliminary knowledge of the radical pair. In the present paper we confine the discussion to fairly general regularities. Since the S-To transition frequency is unique for the RP spin subensemble, in each of them the singlet-triplet evolution proceeds with different effectiveness. Hence, the situation is possible when for subensemble {m’ } the condition (w;“’ 72 B 1) is met, while for another subensemble {m ))} the opposite inequality ( w;lM7, CK 1) holds. It means that different subensembles recombine under different regimes. In such a case it is not easy to obtain reliable information on the recombination process, thus, it is desirable to “move” all subensembles into one and the same recombination regime by choosing the suitable solution viscosity. Below we shall assume that all RP spin subensembles recombine within the same regime. The total RP recombination probability as a switching time function may be presented in the form p(&l)=

c YmPm(to)=

m

c

m

Ym~om[l-An(

1

where yrn is a statistical weight of the RP subensemble tion can be measured by experiment: L?LP(to)=~(to)-WO=-

C ~,wo,f,(t,) m

(58) with nuclear spin configuration

>

where wo= C, ymwOmis the total RP recombination probability gime, when oT7, B 1, using eqs. (44) and (55) we obtain Ap(t,)=-(3Zr,)-‘exp(-3to/4r2)

(m}. The following func-

C ~~(l/~Y-l/@)sin(o~t~). m

(59) in the field B2. For the first recombination

re-

(60)

Here w;l and WY are the S-To transition frequencies in the fields B, and &, correspondingly. The function (60) is a set of damped oscillations with different frequencies and amplitudes. Therefore, the frequencies o;? can only be estimated, not determined. However, the damping time of Ap( to) is a measurable quantity, because it

S.A. Mkhadov et al /Chemcal Physm 166 (1992) 35-49

46

‘;; .$ 2.05 lo-

f L 00

-

2 E o-103 fj

-2.0

R,.,_

-

,““,““,““,““I”“,IIII,““,“l’,Irrmrmnrr,-,

0

25

50

75

100

125

150

swttching

175

200

delay

225

250

275

(US)

3””

Ftg. 1. Manifestation of spm dynamics in the recombination product of a RP havmg no magnetic nuclei in the swttched magnetic field. The figure presents the dependence x(ta)= [p(t,) -p,] /(pz-pI). p(to) ISthe recombmation probability of the RP in the switched field; pI and pz are recombmatton probabilities of this RP m steady magnetic fields B, and &, respectively. The calculation parameters are: Ag=O.O05, T, = I X lo-“’ s, r,=2x lo-‘s, II,=3000 Oe and &=3300 Oe. x(to) is calculated for different values of n (total number of RP partners encounters): (-) n=3; (---) n=5; (---) n= 10; (+)n= 100.

is the same for all oscillations. So, one can measure the time interval between re-encounters, rZ. For the second recombination regime, when o;l7, < 1 and 0;” 72 /& >> 1, the following equation from eqs. (44) and (56):

is obtained

(61) It is a difficult task to obtain information on the kinetic or magnetic resonance If RP parameters meet the condition O?rZsz/&<< 1 (third regime), then

parameters

of the RP in this case.

(62) It is seen from the last formula that the RP dissociation time 7, is a quantity that can be measured by experiment. Similar information on the RP recombination kinetics can be given by CIDNP experiments carried out in a switched magnetic field. It should be emphasized, that CIDNP experiments have an important advantage because of their high sensitivity (it is known that slight variations of RP spin dynamics do not alter the total recombination probability to a considerable extent, but in CIDNP effects these variations may be clearly pronounced). Figs. l-3 illustrate the above discussion. In plotting them we made use of formula (5 1) for different RP parameters given in the captions.

6. Extraction of information from time-resolution experiments. Comparison of informativity of two kinds of experiments The information on RP recombination kinetics can be directly obtained from the time-resolution experiments proceeded in a steady magnetic field. Such kind of experiments are well known [ 12- 16 1, and in the present paper we compare the information of ordinary time-resolution experiments and those with field switching considered above. For that purpose let us derive the formula of the dependence of RP recombination probability on time in a constant external magnetic field. Within the framework of the two-positional model the latter has already been discussed in the literature [ lo,16 1. But here we obtain the formula required making use of our formalism. Substituting 6= w, - w2 = 0 (magnetic field does not change) into eq. (39 ) we obtain

x g(s) [g(s+&) g(s) [g(s+iol)

+g(s-iwi)] +g(s-iwi)]

-2g(s+io,)g(s-io,) +2g(s+io,)g(s-io,)

.

(63)

S.A. Mikhailov et al. / Chewcal Physw 166 (1992) 35-49

60

b

30

3

20

-_

I

fi

2.0

z

00:

i

-2.o-

m

c) ‘0 v 0 00 r: w-1 0 .p2

% $

0

2

A-30 s

-

-4.0

-

-6.0

-

,I ,,,,,,, 0 25

E -4.0 -5.0

50

75

100

~

125

150

is P ti

---

20:

c 0

n=3

~

7

----

n = n=

-

200

delay

-

rl = 5

-----*rclru

10 100

(4

,,),(,,),(,,’ 175

switching

3.0

n=3

-

n-3

225

250

275

300

(YS)

n=3 n = 5 n = 10 n=lOO

l.O-

tl oow z aJ -1.0

-

E? s-20-

:

2 ;

;

-3.0 -4o-

_.,,~,,,.,,,,,,,.,.,,.~....,....,....,...,I

0

25

50

75

100

125

swrtching

(b) 150

175

200

delay

225

250

275

03

300

030

(VS)

swrtching

Fig. 2. Dependence of the recombination probability and the nuclear spin polarization upon the time of magnetic field switching, to,in the recombination product of a RP having one nucleus with spin I=1/2. Plots of (a) x(to) (see fig. 1) and (b) K(&,)= [ u( to)-c, ] / ( o2- u, ). CT( to) is the nuclear spm polarization in a switched field; 0, and Q m steady magnetic fields of amplitudes B, = 1000 Oe and &= 1100 Oe. RP parameters: Ag=O.OOS,A= 20 e, r, =O. 1 ns, r,= 300 ns. The number of total encounters of the radicals of the pair is (-) n=3; (---) n=5; (---) n=lO; (-) n= 100.

And using eq. (40) as the definition

delay

(us)

Fig. 3. The plots of x( lo) and I dependences for a RP with one proton. Unhke the previous figure in the present one there are calculatrons for magnetic fields of B, = 400 Oe and BZ = 440 Oe. The quantum beatings are explicrtly displayed here. These beatings arise because S-To transition frequencies in different RP subensembles are almost equal (----_) n=3; (---) n=5; (---) n= 10; (t-) n= 100.

of g(s) we determine (64)

ThF !$P SubenseM?Ig transform:

r~$@@?jl)@ion kinetics

IT,(t)

can be obtained

from eq. (64) by the inverse

Lap&e

S.A. Mkharlov et al. /Chemrcal Physm 166 (1992) 35-49

48

Writing eq. (64) we proposed that the inequality sr, -=K1 holds, i.e. the function pm(t) describes the recombination process at the time longer than one contact interval, t> rr. The denominator in eq. (64) is a third degree polynomial R,(s) similar to R, (u) of eq. (46 ). Thus, we can express & ( t ) in terms of the parameters a, b and 0 (es. (50)) ~~(~)=Wom(l-Ll(~)) J;,(f)=

1

u2+02 exp(

(u-b)2+w2

-bt)

-exp(

-at)

s

cos(w,) +

(65)

where w,, is determined by eq. (44). The function f(t) introduced here is the analogue of f( to) (eq. ( 5 1) ). And now we discuss what information we can get from time-resolved experiments. In the first place let us consider the RP without magnetic nuclei. As we have already pointed out (see section 5 ), provided n X- 1, there exist three recombination regimes corresponding to different effectiveness of singlettriplet transitions. Within the first regime, when w1r2 s 1, the functionf( t) takes the form 3(fo=exp(-t/2r2)+(W1r,)-‘exp(-3?/4r,)sin(o,t).

(66)

The first term of (66) exceeds considerably the second one, and the 3( t) behaviour is, therefore, basically governed by exp( -t/2?,). Thus, if 3( t) is measured experimentally, the time between re-encounters r2. can be determined. Besides, if the amplitude of the second term is not very small, the S-To transition frequency wi can also be measured. It has been shown above (section 5 ) that the same information can be obtained from switched field experiments for this recombination regime. The second regime is characterized by RP parameters such that W,T, << 1 and o, r,/& >> 1. In this case 3(t) =exp( -o:t,t/2)

.

The parameter w:t2 can be determined from experimentalf( t) measurements, but simultaneous of both RP parameters is not possible. Considering the third regime of RP recombination (w,r,/& << 1 ), we find that

3tf)==w(-tlrc).

(67) determination

(68)

Hence, within this regime the in-cage lifetime of a radical pair, TV,,is a measurable parameter. Thus, for the simplest RP having no magnetic nuclei time-resolved experiments on RP recombination in a steady magnetic field give the same information as experiments in a switched field. This assertion remains valid also for radical pairs with magnetic nuclei. Indeed, representing the total RP recombination probability like expansion ( 58 ) we can easily arrive at the conclusions drawn above.

7. Conclusion The discussion in the last two sections shows that more information can be provided from radical recombination experiments in a switched magnetic field than from experiments in a steady field. Studying the RP recombination in a switched field one can find magnetic resonance parameters of radicals as well as kinetic parameters of RP. As a matter of fact these experiments contain the same information as time-resolved ones. Thus, in this paper we actually propose one more method for RP recombination kinetics investigation which is alternative, to a certain extent, to the known time-resolved methods [ 12- 161. This new method will perhaps possess certain advantages over the methods based on time-resolved experiments. Firstly, one can often extract easier information from switched field experiments than from time-resolved ones. To explain this claim let us refer to the above formulae. It has been shown that f( to) and 3( t) are

S.A. Mikhailov et al /Chemcal Physm 166 (1992) 35-49

functions directly measured in the experiments. Under the appointed can be represented in the form (see eqs. ( 5 5 ) and ( 66 ) )

relation of RP parameters

49

these functions

f(to)x(w,t,)-‘(l--,/02)exp(-ato)sin(olto), 3(fosexp(-t/2r,)+(o,r,)-‘exp(-3t/4r2)sin(o,t). It is clear that the quantum oscillations (second term off(t) ) may be hidden by the first term, which basically determines the total product yield of the radical reaction. The functionf( t,,) does not have that “hiding” term, and quantum beatings are explicitly displayed here. The second feasible advantage of the method proposed is the resolution. The latter of the known time-resolved methods is in the order of a few tens of nanoseconds [ 12- 16 1. The resolution of this new method is defined by the time interval of the “steady” field switching and, it should seem, higher resolution can be achieved by application of this method.

References [ 11 A.B. Doktorov, S.A. Mikhailov and P.A. Purtov, Chem. Phys. 160 ( 1992) 223,239. [2] N.L. Lavnk and V.E. Kbmelinsky, Chem. Phys. Letters 140 (1987) 582. [ 31 N.L. Lavrik and V.E. Kbmelinsky, Khlm. Fiz. 7 ( 1988) 240. [4] K.M. Salikhov, Yu.N. Molin, R.Z. Sagdeev and A.L. Buchachenko, Spin polarization and magnetic effects in radical reactions ( Akadtmia Kiad6, Budapest, 1984). [ 5] M.B. Taraban, V.I. Maryasova, T.V. Leshina, L.I. Rybin, D.V. Gendin and N.S. Vyazankm, J. Organomet. Chem. 326 ( 1987) 347. [ 6 ] O.A. Anisimov, V.M. Grigoryants, V.K. Molchanov and Yu.N. Molin, Chem. Phys. Letters 66 ( 1979) 265. [7] P. Sibani and J.B. Pedersen, Phys. Rev. B 26 (1982) 2584. [ 8 ] J.B. Pedersen, J. Chem. Phys. 67 ( 1977) 4097. [ 91 K.M. Salikhov and S.A. Mikhailov, Teor. Eksp. Khim. 19 ( 1983) 550. [lo] K. Ltlders and K.M. Salikhov, Chem. Phys. 117 (1987) 113. [ 111 G. Kom and T. Kom, Mathematical handbook (McGraw-Hill, New York, 1968). [ 121 R.J. Miller and G.L. Closs, Rev. Sci. Instr. 52 (1981) 1876. [ 131 N.J. Turro, M.B. Zimmt and I.R. Gould, J. Am. Chem. Sot. 105 (1983) 6347. [ 141 M. Laufer and H. Dreeskamp, J. Magn. Reson. 60 (1984) 357. [ 151 H. Fisher, Chem. Phys. 108 (1986) 365. [ 161 Yu.P. Tsentalovich, A.V. Yurkovskaya, R.Z. Sagdeev, A.A. Obynochny, P.A. Purtov and A.A. Shargorodsky, Chem. Phys. 139 ( 1989) 307.