## 3 Results and Discussion
An increase in the value of \(\Delta E_{g}\) may mean that the material becomes less conductive and requires more energy to excite electrons in the conduction band. The widths of the forbidden regions for LaK\({}_{2}\)NiMnO\({}_{5}\) are calculated in the same way. For the LaK\({}_{2}\)NiMnO\({}_{5}\) compound, in the range of 293-403 K, the band gap is 0.96 eV; at 443-463 K, the band gap increases sharply to 4.17 eV. A significant increase in the band gap during the transition to a higher temperature range may be associated with profound changes in the structure of the material, such as the transition from a conductive state to an insulating one. This may be the result of a change in the electronic structure, the formation of new phases, or other significant structural changes. It should be noted that the literature often states that the activation energy of conductivity is equal in magnitude to the bandgap width. However, by definition, the activation energy represents the minimum energy required to excite charge carriers (e.g., electrons or holes) for conduction. This value describes the energy barrier from the valence band to the conduction band and directly influences the material's electrical conductivity. On the other hand, the bandgap width (\(E_{g}\)) refers to the energy difference between the valence band and the conduction band in semiconductors. It determines whether a material behaves as an insulator, semiconductor, or conductor under different temperatures and conditions. When impurity atoms in the crystal lattice substitute for the atoms of the main component, the nature of electrical conductivity changes. The activation energy of conductivity (\(\Delta E_{a}\)) becomes smaller than the bandgap width of the semiconductor (\(\Delta E_{g}\)). In this case, the impurity atom is Ni. This observation aligns with the obtained data: \(\Delta E_{a}\) NdNa\({}_{2}\)NiMnO\({}_{5}\) (0.382 eV) \(<\Delta E_{g}\) NdNa\({}_{2}\)NiMnO\({}_{5}\) (0.87 eV) \(\Delta T\) = 293-353 K; \(\Delta E_{a}\) NdNa\({}_{2}\)NiMnO\({}_{5}\) (0.336 eV) \(<\Delta E_{g}\) NdNa\({}_{2}\)NiMnO\({}_{5}\) (1.53 eV) \(\Delta T\) = 423-483 K; \(\Delta E_{a}\) NdK\({}_{2}\)NiMnO\({}_{5}\) (0.685 \(\circ\) B) \(<\Delta E_{g}\) NdK\({}_{2}\)NiMnO\({}_{5}\) (0.96 eV) \(\Delta T\) = 293-403 K; \(\Delta E_{a}\) NdK\({}_{2}\)NiMnO\({}_{5}\) (0.195 eV) \(<\Delta E_{g}\) NdK\({}_{2}\)NiMnO\({}_{5}\) (4.17 eV) \(\Delta T\) = 443-463 K.  
Let us consider the possible mechanism of conductivity when manganese atoms (trivalent, Mn\({}^{3+}\)) are partially substituted by nickel atoms (divalent, Ni\({}^{2+}\)), which have one less valence electron. The mechanism of excitation of electrical conductivity in this case differs from that of an n-type semiconductor. Nickel atoms form two covalent bonds with neighboring manganese (III) atoms using their own electrons. However, the third covalent bond cannot form in the unexcited crystal due to the absence of a stable electronic configuration for nickel within the lattice. Upon heating, the bonds between manganese atoms weaken, allowing an electron to transfer from one of the neighboring manganese atoms to the nickel atom, thereby forming a third covalent bond. The energy (\(\Delta E_{a}\)) required for this transition is much lower than the energy (\(\Delta E_{g}\)) needed to promote an electron into the conduction band. As the electron moves to the nickel atom, the nickel atom becomes negatively charged, while a hole forms near the manganese atom from which the electron departed. This impurity (Ni) is referred to as an acceptor. The energy level of the acceptor atom is located within the forbidden energy gap, near the top of the valence band, and is separated from the valence band by an activation energy gap (acceptor activation energy). The excitation of electrical conductivity is associated with the capture of a manganese (III) valence electron by the acceptor impurity Ni (II), resulting in the formation of a hole in the valence band. In this scenario, no electrons are present in the conduction band.