Magnetic isotope effect in the presence of external magnetic fields
In this chapter we will discuss one of the MIE’s most remarkable features: its dependence on constant and alternating external magnetic fields. The influence of the external magnetic fields on the isotope effect parameters suggests that MIE operates. The
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Presence of external magnetic fields
90
RPs have triplet precursor molecules and that they are able to recombine only from their singlet state. For such a situation the RP recombination probability increases with an increase of S-T mixing efficiency, since the intersystem S-T crossing pumps RPs from the non-reactive triplet state to the reactive singlet state. 4.1.1 Anisotropic hyperfine interaction (the so-called relaxation mechanism of singlet-triplet transitions) We start with discussing the following model situation. Assume that RPs have triplet precursors, while they can recombine only from their singlet state. Further we assume that interradical spin-spin interactions (Heisenberg exchange and dipole-dipole) can be neglected when two partners of a radical pair are well separated, i.e., in time intervals between two subsequent re-encounters of radicals. Under these circumstances, if the anisotropic hfi is the predominant contributor to S-T transitions, then the RP recombination probability has to decrease with an increase in external field intensity. The anisotropic hfi is modulated by the random rotational motion of radicals, and its contribution to the spin evolution of the RP unpaired electrons at least in non-viscous solvents can be described in terms of the paramagnetic relaxation times Tl and T2 • According to Eqs. (2.56) the rates of the hfi induced paramagnetic relaxation diminish with an increase in the field intensity Bo. The kinetic equations describing the time variation of the population of RPs in singlet and triplet states, caused by the paramagnetic relaxation, are [5, 34, 45, 46]
t -(
(~ ~ 4~' + 2~}SS + (- 4~' + 2~}T'T' +
(a::J
ToTo
4~' (PT+]T+] 1
= (+
+ PL]L]) ,
4~' + 2~,JpSS - (4~' + 2~,JPTOTO 1
2
4~' (PT+]T+] 1
1
2
+ PL]L]) ,
1- ( ) 11 - 41:' Pss + PToTo - 21:' PL]L] +--Re 21:" PSTo ' 1
1
1
4.1 Constant fields
( a::J
STo
91
==
-(4~' + 2~,JpSTO +(- 4~' + 2~,JPTOS 1
2
1
2
(4.1)
- 4 ;" (PT+1T+1 - PL1L1) , 1
where 1
1
1
1
-==-+1;.' 1;. A 1;.B
1
1
1
1;B
T," 1
-==-+1;'
T2A
1
1
==---
where I/T1A' lITlB , lIT2A , I/T2B are the rates of the longitudinal (111'1) and the transverse (1/1'2) relaxation of radicals A and B of a pair, respectively. The contribution of each magnetic nucleus to the paramagnetic relaxation rates is given by the following expressions (see also Eqs. (2.56)) 1
2W
==----1;. 1 + (Ye BoTO)2 '
~ == W(1 + 1 1; 1 + (YeBOTO)
2
J'
(4.2)
where W = (2/3)/(/ + 1)(reYnli)2r-6To, r is the distance between the electron and the nucleus, To is the correlation time of the radical's rotational motion, and Bo is the intensity of the external magnetic field. In the presence of many magnetic nuclei, their contributions to the relaxation rates are added (see, e.g., Eq. (3.33)). From Eqs. (4.2) it follows that 111'1 tends towards zero at high magnetic field intensities. Thus, in the high field limit the kinetic Eqs. (4.1) reduce to
==0, ==0 1
1
== - 2 1',. ' PSTo + 21',. ' PToS 2
2
(4.3)
92
Presence of external magnetic fields
These equation
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