Origin of magnetic moments and ferromagnetic properties of potassium clusters in zeolite A

K clusters arrayed in zeolite A are investigated in detail. K clusters are generated in regular alpha-cages of zeolite A by the loading of guest K metal at a loading density of K atoms per alpha-cage, $n$. The value of $n$ was changed from 0 to 7.2. It is known that this system shows ferromagnetic properties for $n > 2$, where the Curie temperature increases with $n$, has a peak of approx,8 K at $n approx 3.6$, and decreases to 0 K at $n = 7.2$. The negative Weiss temperature is estimated from the Curie-Weiss law for $n > 2$. A spherical quantum-well (SQW) model for the K cluster with 1$s$, 1$p$, and 1$d$ quantum states has previously been proposed, and the ferromagnetic properties were explained as being due to $s$-electrons in 1$p$ states. A spin-cant model of Mott-insulator antiferromagnetism in a K cluster array has been proposed for the origin of the ferromagnetic properties. However, the SQW model cannot explain either the $n$-dependence of the Curie temperature or that of the Curie constant. In the present study, detailed measurements were made of the optical reflection, magnetic properties, electron spin resonance, and electrical resistivities. An optical reflection band at 0.7 eV is clearly observed for $2 < n < 6$ at low temperatures, and is assigned to the excitation of the sigma-bonding state between 1$p$-states in adjoining alpha-cages. The electrical resistivity indicates an insulating state continuously for $n$. We propose an advanced SQW model with consideration for sigma-bonding, the orbital orthogonality of 1$p$-hole states, and also the superlattice structure. We explain the observed electronic properties using this model based on a correlated polaron system. We propose an enhancement effect of the Dzyaloshinsky-Moriya (DM) interaction between adjoining $1p$ orbitals by the extension of the Rashba mechanism of spin-orbit interaction.