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We have used time-of-flight inelastic neutron scattering to measure the spinwave spectrum of the canonical half-doped manganitePr$_{0.5}$Ca$_{0.5}$MnO$_{3}$, in its magnetic and orbitally ordered phase. Thedata, which cover multiple Brillouin zones and the entire energy range of theexcitations, are compared with several different models that are all consistentwith the CE-type magnetic order, but arise through different exchange couplingschemes. The Goodenough model, i.e. an ordered state comprising strong nearestneighbor ferromagnetic interactions along zig-zag chains with antiferromagneticinter-chain coupling, provides the best description of the data, provided thatfurther neighbor interactions along the chains are included. We are able torule out a coupling scheme involving formation of strongly bound ferromagneticdimers, i.e. Zener polarons, on the basis of gross features of the observedspin wave spectrum. A model with weaker dimerization reproduces the observeddispersion but can be ruled out on the basis of discrepancies between thecalculated and observed structure factors at certain positions in reciprocalspace. Adding further neighbor interactions results in almost no dimerization,i.e. recovery of the Goodenough model. These results are consistent withtheoretical analysis of the degenerate double exchange model for half-doping,and provide a recipe for how to interpret future measurements away fromhalf-doping, where degenerate double exchange models predict more complexground states.

Spin excitations used to probe the nature of the exchange coupling in the magnetically ordered ground state of Pr$_{0.5}$Ca$_{0.5}$MnO$_{3}$