Magnetic properties and magnetic structure of the Ba$_{2}$Mn(PO$_{4}$)$_{2}$ antiferromagnet featuring frustrated zigzag chains of $S=frac{5}{2}$ Mn$^{2+}$ ions are reported based on neutron diffraction, density-functional band-structure calculations, as well as temperature- and field-dependent measurements of the magnetization and specific heat. A magnetic transition at $T_Nsimeq 5$,K marks the onset of the antiferromagnetic order with the propagation vector ${mathbf k} = (frac12, 0, frac12)$ and ordered moment of $4.33pm0.08~mu_B$/Mn$^{2+}$ at 1.5,K, pointing along the $c$ direction. Direction of the magnetic moment is chosen by the single-ion anisotropy, which is relatively weak compared to the isostructural Ni$^{2+}$ compound. Geometrical frustration has strong impact on thermodynamic properties of Ba$_2$Mn(PO$_4)_2$, but manifestations of the frustration are different from those in Ba$_2$Ni(PO$_4)_2$, where frustration by isotropic exchange couplings is minor, yet strong and competing single-ion anisotropies are present. A spin-flop transition is observed around 2.5,T. The evaluation of the magnetic structure from the ground state via the spin-flop state to the field-polarized ferromagnetic state has been revealed by a comprehensive neutron diffraction study as a function of magnetic field below $T_N$. Finally, a magnetic phase diagram in the $H-T$ plane is obtained.
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