"Goodenough Model" victim of an abuse of language
an INS analysis lie, that hides it in fact discard it!
subtly making an abuse of the itinerant electron picture
of the advocated DDEX theory of D. Khomskii


All Inelastic Neutron Scattering Studies of the CE-state
Advocate a A DDEX model that is NOT Goodenough Model!
Inelastic neutron scattering (INS)  is indeed the RARE if not the ONLY experimental technique that can extract magnetic exchange constants at the microscopic level by a analysis of the spin waves. However during 10 years,  INS work INGNORED the crystallography/symetry, and has never adressed the exchange model that can be deduced from ZP symmetry properly (I insist: a published  work): it has never aknowledged the orbital order suggested by the ZPO, how GKA rules can be adapted, and how additionally, the symmetry can constrain the exchange constant. But even before getting into the detail of this "omission" & replacement by verbosing indigest theoretical references, and totally independently of the validity of ZPO, let us first comment on the way that INS studies claim they confirm the "Goodenough Model", but they are in fact NOT studying/confirming the Original Goodenough Model strictly speaking.

In Braden's group INS studies of single layered, and Perring's subsequent work on bi-layered half-doped manganites and ULTIMATELLY and finally stated EXPLICITELY in the study of 3D PCMO50 of the CMR Pr-Ca system,  data are fit by a modified "Goodenough Model justified by DDEX"  shown in Fig 1a of the 3D PCMO50 paper (which should rather be threrefore sipmly called "DDEX Model"). It is a model copied by Perring  from Braden's group : this model is adding long range interactions (J2 and J3) in the INS model they advocate and ascribed to the Goodenough model (which strictly speakning, consist of 1st neighbor J1 and JAF interaction only.

The CHOICE of these added interactions is partly ARBITRARY (for J3) and the consideration of J2 is implicitely made because of the assumption that Double Exchange, not Goodenough Super exchange, would be building the ferromagnetic zigzag of the CE-type magnetic structure. The phenomenological addition of these two long range in plane interactions adds a level of complexity to the true and Original Goodenough Model that's strictly speaking, a 1st neighbor Heisenberg model with Super Exchange interactions, with a simplicity that makes all its beauty.

This complication is IN FACT meant to capture an observed gap in spin wave spectra in INS that would not b e captured by the Original Goodenough Model alone. With this parametrisation of the exchange, authors get two F interaction (one very strong for the 1st neighbor J1 and one significaant for the long  range J2) seemingly well justified by double exchange along the F-zigzag, BUT (to get the gap) this need to be compensated by an interaction (2nd neighbor, J3), fond to be AF along the zigzag, which contradicts in a way the assumption made in the modelling/understanding :  indeed, if Double exchange prevails, J3 should be also F but neither this nor its physical origin, is discussed. 

The conceptual confusion made between the Goodenough Model and DDEX model becomes pathologically visible in Perring's et al paper on PCMO, and is reflected by the most confusing description/justification the DDEX in the data analysis: "The DDEX model17,18, which is a more general version of double exchange, would be an appropriate starting point for analyzing our data. However, calculation of the excitation spectra within such a [DDEX] framework is, though possible, far from straightforward49,50. In the absence of such calculations, we therefore use an effective Heisenberg Hamiltonian to model our data, just as was done for the half-doped manganites mentioned in the introduction39,41,42.The use of an effective Heisenberg Hamiltonian is a well-established procedure to analyze the spin wave spectra in magnetically ordered systems, and not just those with purely integer or halfinteger spins, nor even solely insulators. The exchange interactions in the model [Heisenberg] Hamiltonian that are required to reproduce the observed dispersion relations and intensities establish the strength, range and symmetries of the magnetic interactions.", followed by examples, concluding "exchange could be qualitatively explained by mapping the excitations of the double exchange model onto an equivalent ferromagnetic Heisenberg Hamiltonian": in other and simpler words, authors say they assume DDEX to be valid but can't analyse data with DDEX so they use effecvtive Heisenberg exchange and linear spin wave theory because it is merely justified by habit!

The problem of such verbosing justification of data analyses of spin waves in manganites, which is to use effective Heisenberg exchange to describe more metallic systems (here, systems assumed to follow the Double exchange model), it is that if the conclusion is that the real & original Goodenough Model is the best, only 1st Neighbor interactions should suffice to model the data, and the whole problem, is that this is not the case!

In the INS prefered model of the CE-state
The NEED to add longer range interaction
inspired by the DDEX implicitely discard
the Goodenough Model because it's meant
to  change fundamentally the nature
of exchange from SE to DE

In truth, what applies to all existing INS analyses of compounds showing the CE-type magnetic structure is that they cannot cope with the model degeneracy of exchange models built on the Charge ordered super cell. SO BEFORE even considering the arbitrarily chosen long range F interaction J2 BEYOND 2nd nearest neighbors to construct the pseudo-Goodenough Model (motivated by the amalgam between Goodenough and DDEX theory), there are many other 2nd nearest neighbor Super-super-exchange pathways such as J2, whose sign is unknown and which structural arguments can hardly constrain as easily, as for 1st neighbors interactions...