# Magnetic Field Effect in Bimolecular Rate Constant of Radical Recombination

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## Abstract

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## 1. Introduction

_{1}relaxation time of the NO radical is very short and constitutes the order of one picosecond. Such a short spin relaxation should give rise to the disappearance of the magnetic effect. Nevertheless, a noticeable magnetic effect is observed because there is a large difference in the g-factors, which leads to a large difference in the Larmor frequencies in magnetic fields of the order of several Tesla, and this competes with relaxation. A theoretical study which considered the $\mathsf{\Delta}g$ mechanism and paramagnetic relaxation [43] was also applied in the same work to interpret the dependence of the rate constant of the bimolecular recombination of the complex of ruthenium radicals with bipyridine $Ru{(bpy)}_{3}^{3+}$ and methyl viologen $M{V}^{+}$ on viscosity and magnetic field. The relaxation time of paramagnetic ruthenium complexes was several dozen picoseconds, and the g-factor difference was also approximately equal. In Refs. [38,41,42], the magnetic effects in the recombination of radicals diffusing on the plane were considered (i.e., the two-dimensional case of radical recombination). In particular, in our work [42], the influence of paramagnetic relaxation on recombination, hyperfine interactions of radical spins with magnetic nuclei, and the difference in the g-factors of radicals were taken into account as well. This model addresses the possible magnetic effects in the recombination of lipid radicals that producer a diffusion motion (lateral diffusion) across the cell membrane surface.

## 2. Results

#### 2.1. Theory

**S**and the triplet state ${T}_{0}$ (singlet–triplet mixing). In the high-field approximation (12), we obtained an analytical expression for the bimolecular radical recombination rate constant, also taking into account the paramagnetic relaxation of the radical spins (see Supplementary material).

#### 2.2. Magnetic Field Effects Calculation Results

## 3. Discussion

## 4. Materials and Methods

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Dependence of the radical recombination rate constant in units of the diffusion-controlled reaction rate constant ${k}_{D}$ (diffusion constant) on the magnetic field. The solid black line is the exact numerical calculation, and the red open circles are the analytical calculation (according to Equations (S2) and (S3) of the Supplementary material) in the high-field approximation. (

**Insertion A**) is the rate constant for the range of very large magnetic fields; there, the high-field approximation totally coincides with the exact calculation. Therefore, dependencies are not distinguishable. (

**Insertion B**) shows the recombination rate constant in a range of small magnetic fields. Model parameters: contact radius $R=10\text{}\mathrm{\u212b}$, relative diffusion coefficient $D=2.0\xb7{10}^{-5}\mathrm{c}{\mathrm{m}}^{2}/\mathrm{s}$, hyperfine coupling constant in the first radical ${a}_{1}=0.5$ mT, hyperfine coupling constant in the second radical ${a}_{2}=1.0$ mT, $g$ -factor of the first radical ${g}_{1}$ = 2.0, $g$-factor of the second radical ${g}_{2}$ = 2.001, $\frac{{k}_{S}}{{k}_{D}}=100$ (diffusion controlled regime). Longitudinal, ${T}_{1A}$, and transverse relaxation times ${T}_{2A}$ of the radical $A$ are: ${T}_{1A}=1000$ ns, ${T}_{2A}=1000$ ns, we neglect by relaxation of $B$ radical.

**Figure 2.**Magnetic field effect (MFE) in the recombination rate constant k(B)/k(B = 0). Insertion—magnetic field effect on the range of small magnetic fields. The other parameters are the same as in Figure 1.

**Figure 3.**MFE in the recombination rate constant for different diffusion coefficients. Red line $D=2.0\xb7{10}^{-8}\frac{\mathrm{c}{\mathrm{m}}^{2}}{\mathrm{s}}$, green line $D=2.0\xb7{10}^{-7}\frac{\mathrm{c}{\mathrm{m}}^{2}}{\mathrm{s}}$, blue line $D=2.0\xb7{10}^{-6}\frac{\mathrm{c}{\mathrm{m}}^{2}}{\mathrm{s}}$, black line $D=2.0\xb7{10}^{-5}\frac{\mathrm{c}{\mathrm{m}}^{2}}{\mathrm{s}}$. The other parameters are the same as in Figure 1.

**Figure 4.**Magnetic field effect k(B)/k(B = 0) for different relaxation times. Black line: ${T}_{1A}=1000\text{}\mathrm{n}\mathrm{s},{T}_{2A}=1000\text{}\mathrm{n}\mathrm{s};$ green line: ${T}_{1A}=1000\text{}\mathrm{n}\mathrm{s},{T}_{2A}=250\text{}\mathrm{n}\mathrm{s}$; red line: ${T}_{1A}=250\text{}\mathrm{n}\mathrm{s},{T}_{2A}=250\text{}\mathrm{n}\mathrm{s}$. The other parameters are the same as in Figure 1.

**Figure 5.**Magnetic field effect $k(B)/k(B=0)$ for equal hyperfine constants ${a}_{1}={a}_{2}=0.5\text{}\mathrm{m}\mathrm{T}$ in radicals. Black line $D=2.0\xb7{10}^{-5}\frac{\mathrm{c}{\mathrm{m}}^{2}}{\mathrm{s}}$, red line—$D=2.0\xb7{10}^{-8}\frac{\mathrm{c}{\mathrm{m}}^{2}}{\mathrm{s}}$. Other parameters are the same as in Figure 1.

**Figure 6.**Magnetic field effect k(B)/k(B = 0) for equal g-factors of radicals, ${g}_{1}={g}_{2}=2.0$ The black line corresponds to $D=2.0\xb7{10}^{-5}\frac{\mathrm{c}{\mathrm{m}}^{2}}{\mathrm{s}}$, the red line corresponds to $D=2.0\xb7{10}^{-8}\frac{\mathrm{c}{\mathrm{m}}^{2}}{\mathrm{s}}$. Other parameters are the same as in Figure 1. Insertion—the magnetic field effect on a smaller scale.

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**MDPI and ACS Style**

Doktorov, A.B.; Lukzen, N.N.
Magnetic Field Effect in Bimolecular Rate Constant of Radical Recombination. *Int. J. Mol. Sci.* **2023**, *24*, 7555.
https://doi.org/10.3390/ijms24087555

**AMA Style**

Doktorov AB, Lukzen NN.
Magnetic Field Effect in Bimolecular Rate Constant of Radical Recombination. *International Journal of Molecular Sciences*. 2023; 24(8):7555.
https://doi.org/10.3390/ijms24087555

**Chicago/Turabian Style**

Doktorov, Alexander B., and Nikita N. Lukzen.
2023. "Magnetic Field Effect in Bimolecular Rate Constant of Radical Recombination" *International Journal of Molecular Sciences* 24, no. 8: 7555.
https://doi.org/10.3390/ijms24087555