Time-Reversal Symmetry in Spin Systems in the Presence and the Absence of Redox Processes

Introduction

The redox systems are chemical systems, in which reduction and oxidation reactions occur (Amman, 2018). In the decomposition redox reaction AB → A+B the products are formed by the decomposition of the reactants. The reaction2H₂O → 2H₂+O₂(1) is an example of decomposition redox reaction. Particularly, the reactionCaCO₃ → CaO + CO₂(2) is a decomposition one, not a redox reaction, since the oxidation state is not changed here (https://biologyonline.com/dictionary/redox-reaction). The combination redox reaction is a process opposite to that in the decomposite reaction. The oxidation state of reactants is increasing at oxidation and decreasing at reduction. The reducing agents are electropositive elements, while the electronegative elements are oxidizing agents. The reactionZn + CuSO₄ → ZnSO₄ + Cu(3) is an example of displacement redox reactions with the oxidation half-reactionZn → Zn²⁺+ 2e^-(4) and the reduction half-reaction

The time-reversal symmetry is one of maim symmetries in physics. Let us consider consider a simple example of displacement redox reactions described by formulas (3) - (5). During the redox reaction the paramagnetic ions Cu²⁺ with the electron configuration 3d⁹ and spin S = 1/2 are transformed into copper atoms with the electron configuration 3d¹⁰4s¹. In this case, the antiunitary time-reversal operator K has the form (Wigner, 1959).K=U^(1/2)K₀, U^(1/2)= i𝜎_y,(6) where U^(1/2) is a unitary 2x2- matrix (U^(1/2)· U^(1/2)+= 1), 𝜎_y is the imaginary Pauli matrix and K₀ is the operator of complex сonjugation. The anti-unitarity of the operator K is determined by the relation K (𝛼𝜓+𝛽𝜑)= 𝛼^*K𝜓+𝛽^*K𝜑, where 𝜓 and 𝜑 are wave functions, while 𝛼 and 𝛽 are complex coefficients.

In this paper, the time-reversal symmetry in spin systems in the presence and absence of the redox processes is studied. In Section 2, a brief discussion of the influence of the redox processes on the time-reversal symmetry is given. In Section 3, the time-reversal symmetry related to Cobalt binuclear compounds with redox-active bridging is studied. In Section 4, the consequences of the decomposition under redox processes of the cluster formed by Mn(II) ions on the time-reversal symmetry of the final reactants are considered. In Section 5, it is performed the transformation of the antiferromagnetic exchange interaction dimer cluster Mn(III) – Mn(III) into exchange interaction of ferromagnetic type in dimer clusters Mn(III) – Co(III). Such a transformation takes place when one of the ions of a two-nuclear cluster is isomorphically replaced by an ion, whose wave functions are reversed in time with respect to the wave functions of a substituted ion. Section 6 contains main conclusions.