Title: Aggregate Frequency Width, Nuclear Hyperfine Coupling and Jahn-Teller Effect of $Cu^{2+}$ Impurity Ion ESR in $SrLaAlO_4$ Dielectric Resonator at $20$ Millikelvin

Abstract: The impurity paramagnetic ion, $Cu^{2+}$ substitutes $Al$ in the $SrLaAlO_4$ single crystal lattice, this results in a $CuO_6$ elongated octahedron, the resulting measured g-factors shows four-fold axes variation condition. The aggregate frequency width of the electron spin resonance with the required minimum level of impurity concentration has been evaluated in single crystal $SrLaAlO_4$ at $20$ millikelvin. Measured parallel hyperfine constants, $A_{\scriptscriptstyle\parallel Cu}$, were determined to be $-155.7\times10^{-4}~cm^{-1},~ -163.0\times10^{-4}~cm^{-1},~ -178.3\times10^{-4}~cm^{-1} $ and$~-211.1\times10^{-4}~cm^{-1}$ at $9.072~GHz~(WGH_{4,1,1})$ for the nuclear magnetic quantum number $M_I=+\frac{3}{2},+\frac{1}{2},-\frac{1}{2}$,~and$-\frac{3}{2}$ respectively. The anisotropy of the hyperfine structure reveals a characteristics of static Jahn-Teller effect. The second-order-anisotropy-term, $\sim (\frac{spin-orbit~coupling}{10D_q})^2$, is significant and can not be disregarded, with the local strain dominating over the observed Zeeman-anisotropy-energy difference. The Bohr electron magneton, $\beta=9.23\times 10^{-24} JT^{-1}$, (within $-0.43\%$ so-called experimental error) has been found using the measured spin-Hamiltonian parameters. Measured nuclear dipolar hyperfine structure parameter $P_{\scriptscriptstyle\parallel}=12.3\times10^{-4}~cm^{-1}$ shows that the mean inverse third power of the electron distance from the nucleus is $\langle r^{-3}_q\rangle\simeq 5.23$ a.u. for $Cu^{2+}$ ion in the substituted $Al^{3+}$ ion site assuming nuclear electric quadruple moment $Q=-0.211$ barn.