Interplay between lattice and magnetism in distorted diamond spin chain compound Cu$_3$Nb$_2$O$_8$

Cu$_3$Nb$_2$O$_8$ is an atypical type-II magnetic multiferroic, where the direction of the electric polarization does not conform with the conventional spin current model. It exhibits a unique crystal structure, where Cu atoms arrange themselves into a distorted diamond chain along the crystallographic $a$-axis, leading to quasi-one-dimensional behavior with the presence of nonmagnetic Nb atoms between these chains. To understand its multiferroic character, we employed a combination of temperature-dependent powder X-ray diffraction, X-ray absorption spectroscopy, and magnetization measurements. Anomaly in lattice parameters is observed below the magnetic transition temperature (T$_N$), and the low temperature crystal structure lacks inversions symmetry. The near edge of the X-ray absorption spectra confirms 2+ valence state of Cu, which remains unaltered down to 20 K. Extended X-ray absorption fine structure analysis reveals significant alterations in the Cu–O bond lengths within the first shell subsequent to T$_N$. The pre-edge intensity increases sharply below T$_N$, which justifies the increase in local distortion of the Cu–O coordination. Notably, our analysis of magnetic susceptibility data indicate stronger interaction between the shorter diagonal Cu atoms in the diamond chain than that of predicted by Density Functional Theory. The work sheds new light on the interplay between structure and magnetism in CuNbCu$_3$Nb$_2$O$_8$ and it contributes to understand this multiferroic materials.