Full propagation-vector star antiferromagnetic order in quantum spin trimer system ${\rm Ca_{3}CuNi_2(PO_4)_4}$

We show that the antiferromagnetic structure in the quantum spin trimer system ${{\rm Ca_{3}CuNi_2(PO_4)_4}}$ is based on both arms of propagation vector $\vec{k}$ star $\{[{1\over2},{1\over2},0],[-{1\over2},{1\over2},0]\}$ of the paramagnetic space group $C2/c$. The structure is generated by a symmetric direction of the order parameter of two dimensional irreducible representation of $C2/c$ with one active magnetic mode and corresponds to the Shubnikov magnetic space group $C_a2/c$. We reveal the relation between representation analysis in the propagation vector formalism and Shubnikov symmetry. These types of multi-$\vec{k}$ structures are extremely rarely observed experimentally. To further prove the specific magnetic structure we have performed the calculations of the spin expectation values in the isolated Ni$^{2+}$-Cu$^{2+}$-Ni$^{2+}$ trimer with realistic Hamiltonian. The calculated spin values $<\!S_{\rm Ni}\!\!>={0.9}$ and $<\!S_{\rm Cu}\!\!>={0.3}$ are within 10% accuracy in agreement with the experiment, providing strong complimentary argument in favor of multi-arm magnetic structure.