Our magnetic exchange associated to the ZP model (cited less than 10 times...), is DEFINING the THIRD Heuristic Heisenberg exchange model adapting the GKA rules to ZPO Oribital order symetry. It is the alternative model INS should test but has never tested, in addition to:
 
1) the historically accepeted and also heuristic/semi-empirical ionic Mn3+/Mn4+ charge ordering (CO) scenario (Goodenough) - a localised electron picture, in which a Heisenberg model of 1st neighbor anisotropic Super exchage interactions depending on localised eg orbital configurations (the GKA or Goodenough Kanamori Anderson rules) a priori suffice explain the peculier CE-type magnetic order. Experimentally, the Goodenough Model is justified by the octahedral distortions determined in the seminal and pionering investigations, by Radaelli

and

2) modern theoretical topological band insulator (TBI) scenario promotted by the doubly-degenerated double exchange model - (D)DDEX (See Dagotto, for a review, and Effremov/Khomski, for the specific case of Pr1/2Ca1/2MnO3 - PCMO system), which is an itinerant electron picture, in which eg valence electrons are delocalised by the Double exchange mechanism along the ferromagnetic zigzag of the CE-type structure and only inner t2g electron are coupled by an isotropic Super exchange: DDEX is NOT a Heisenberg model, but a Hamiltonian setting in competition the Heisenberg exchange of t2g spins and the Double Exchange of eg spins coupled by strong Hund rule.

Let us detail it more: the definition of DDEX is part of the a search of minimal models for the competing phases of manganites, in the context CMR was phenomenologically understood as Phase Separation. I quote from Dagotto:

"Although there is little doubt that the famous CEstate of Goodenough, reviewed in Sec.III.a, is indeed the ground state of x=0.5 intermediate and low bandwidth manganites, only very recently such a state has received theoretical confirmation using unbiased techniques, at least within some models. In the early approach of Goodenough it was assumed that the charge was distributed in a checkerboard pattern, upon which spin and orbital order was found. But it would be desirable to obtain the CE-state based entirely upon a more fundamental theoretical analysis, as the true state of minimum energy of a well-defined and realistic Hamiltonian. If such a calculation can be done, as a bonus one would find out which states compete with the CE-state in parameter space, an issue very important in view of the mixed-phase tendencies of Mn-oxides, which cannot be handled within the approach of Goodenough."

p36 "Topological Origin of the x=0.5 CE-state
The fact that λ does not play the most crucial role for the CE-state also emerges from the “topological” arguments of Hotta et al. (2000), where at least the formation of zigzag ferromagnetic chains with antiferromagnetic interchain coupling, emerges directly for λ=0 and large JH, as a consequence of the “band-insulator”character of those chains. Similar conclusions as those reached
by Hotta et al. (2000), were independently obtained by Solovyev and Terakura (1999), Solovyev (2000), and by van den Brink et al. (1999). The concept of a band insulator in this context was first described in Hotta et al. (1998)."

concluding p36 "In summary, at x=0.5, the CE-type AFM phase can be stabilized even without the Coulombic and/or the JT phononic interactions, only with large Hund and finite JAF couplings. [...] Thus the complex spin-charge-orbital structure for half-doped manganites can be understood intuitively simply from the viewpoint of its band-insulating nature"

From there, an amalgam can be easily made, as both the original Goodenough model and the modern DDEX approach provide two alternative model of the CE-type magnetic order having similar (yet not exactly the same) orbital order; the DDEX view point of the CE-state is compatible with Radaelli's symetry, but it is NOT compatible with the refined distortion of Radaelli's structure which justifies the ionic Goodenough model:

By design, in "pure" DDEX models, the Charge Order is CONCEPTUALLY different from the ionic order assumed by the Goodenough Model BECAUSE, the DDEX and Goodenough pictures of the CE state reflect in reality two opposite conceptions of Strongly correlated electron systems, corresponding to fully localised and fully itinerant description of the electronic structure,  respectively associated to mutually exclusive Goodenough Super Exchange  and  Double Exchange intercations as two different if not opposite ways to explain the same CE-type magnetic order.

In the TBI scenario, double exchange via degenerate eg orbitals drives the electronic structure to form a Topological Band Insulator having a mixed valent electronic structure for Mn atoms. The eg electorn form 1D zigzag chain that are "band-insulating" and those electronic bands project in a way that a dz2 orbital character appear on the nominal Mn3+ bridge site and no orbital character on nominal Mn4+ corner sites of the original Goodenough Model in which very differently, the Mn3+ site is assumed to have a full dz2 ocupation. The structure has P21/m symetry but distortions are different from Radaelli's model so that the 3 Mn site have same intermediate valence. This is a model generally advocated by RXS.

The CE-type magnetic structure is captured by the DDEX model using Double Exchange interaction in doubly degenerate eg levels INSTEAD OF Super exchange interactions of localised eg valence electrons whose sign is semi-empirically given by the GKA rules... In DDEX, an isotropic super exchange is only applied to core t2g electrons. 


which is exactly what Toby Perring's INS studies of the PCMO ground state will allow himself to do, mixing two antagonist concepts, one from his choice of DDEX as guide, and my endless calls to make GKA rules/OO considered more seriously.
 

ISIS+Oxford-INS made forever, a Conceptual Mistake regarding the CE state Phenomenology/theory in PCMO manganites, by making advocating a DDEX theory and calling it "Goodenough model": the conceptually flaw  is more apparent in a subtle variant of the Abstract of their last production, (3D PCMO50),  which you can quote from the Abstract of this Proceeding p 36 (the only public advertisement to my knowledge that's listing and gathering all collaborators of Toby Perring including those who published the first overrated discard of the ZP model:


  This is Abstract hides in fact two  conceptual flaw: first, the sentence "GKA rules recieve support from DDEX" is epistemically a WRONG understanding of the Subject. This will root the "abuse of language" of the "Goodenough Model" wording : and second, it is not true that DDEX "predict" more complex ground states, we will clarify, this sentence refers to ground states that are "imposed", not "predicted" by the advocated theory, hindering the fact that this theory makes assumptions that are actually phenomenologically wrong and particulary misleading, for INS studies!


Recalling that... it is actually our KEY idea : the Goodenough Model adapted exchange model to the ZPO proposal that ISIS-INS blanks out! that raised the interest of Daniel Khomski when I was meeting him in Conference, and who got inspired by the non-colinear magnetic order I was proposing, and the non-centrosymetry of the experimental structure I determind to propose his theory.

Khomski et al used a DDEX version of our proposal (called Bond centered CO - BC-CO having an "orthogonal" magnetic phase) assumed at x=0.4 and a DDEX version of the Goodenough model (the site centered SC-CO that assumes CE-type order) assumed at x=0.5 and suggested that their mixture in "intermediate" phases (0.4<x<0.5) is multiferoic, thus, defining a concept of Multiferroicity induced by charge ordering, which induced very fruitful reasearches, especially in the PCMO system.

However, one assumption made by Khomskii et al DDEX theory of PCMO is certainly phenomenologically wrong : PCMO40 has a so called pseudo-CE magnetic order, not an orthogonal order : the theoretical work assumes below x=0.5 IRREALISTIC magnetic structures. These magnetic structures all have in plane spin order going from collinear CE-type, to orthogonal arrangement of dimers through intermediate phases by continuous rotation. It is therefore extremely important to admit that


It explicitely states that INS can/will prove the ground state "predicted" by DDEX at x=0.4 "Away from half-doping" which is a ZPO with the orthogonal magnetic order that WE originally PROPOSED... as potential alternative to the CE-structure at half doping, not at x=0.4...

The problem being that that orthogonal phase has no reality... and has therefore nothing to do with the experimental magnetic structure of the PCMO system which are goining from in plane CE-type to pseudo CE-type, with spins oriented out of plane - see Jirak, going from x=0.5 to x=0.4 !!!


The assumption permits an apparent pseudo agreement with the new STRUCTURAL phenomenology experimentally brought by the ZP odering proposal. But it is obtained by a misconsideration of the real/true phenomenology of the magnetic order. And this is acceptable, because the real purpose of the article is clearly stated : it is to define a NEW CONCEPT : mulriferoicity induced by charge ordering. By its assumptions (not using the experimental pseudo-CE type magnetic order at x=0.4 ), it was not intending to elucidate the details of the PCMO ground state.

The INS support for that "pure" DDEX theory is not rendering justice to it, by stating that it "predicts more complex ground states" away from half-doping. The use of the word "predict" is comppletely unjustified and untrue, not admitting that the DDEX theory is in fact not predicting but has decided to assume (at x=0.5) and make up (at x=0.4, inspired by ZPO) ground states for its purpose: the definition of the "mulriferoicity induced by charge ordering" CONCEPT NOT Intended to bring a better understanding magnetism in PCMO.

It is a shame  that this DDEX work has been proposed without stating more explicitely it's at the expense of a rigourous consideration of the historical/phenomenological understanding of the PCMO MAGNETIC phase diagram/REAL MAGNETIC STRUCTURES much straighforwardly understood with the ionic Goodenough Model. Because I insist this is not the DDEX theory's fault

But what's stated as intent in the INS 3D PCMO50 study is NOT what is achieved in the conclusion, which claim it is distinguishing/discriminating between Goodenouh Model and the ZPO model,  I quote  ; "It is our intention that this detailed description of how inelastic neutron scattering data can be analyzed to distinguish between the different scenarios that can arise in DDEX models will serve as a reference for future work on manganites away from half-doping [towards x=0.4], for which the calculations indicate a greater instability towards the formation of dimer-like ground states."; That intent is to distinguish between the SC-CO and BC-CO DDEX model : it is deliberately RESTRICTING the discussion to a BAND picture which is ITSLEF RESTRICTING the BC-CO linked to ZPO at x=0.4.

This has not only lead to wrongly take what's advocated (the SC-CO phase modelling the CE-type structure of the DDEX theory), as a justification for the Goodenough model, but overrate a theory that just cannot be relied upon to adress the magnetism/magnetic excitations of PCMO..  The seriousness of the conceptual mistake, is hindered by the impact factor of the DDEX work of Effremov/Khomski cited more than our ZPO proposal that inspired it... and the the way it is used by INS to circumvent the "crystallographic controversy".

This leads to confusingly antagonise TWO MODEL INSTEAD OF THREE associated for INS to the two CONTROVERSIALLY DISPUTED symetries, one NAMED AFTER (but it is not) the strictly speaking the TRUE Goodenough picture and the other NAMED AFTER (but not quite close to...) the ZP model, which are both localised electron models recuperated within THE THIRD itinerant DDEX Effremov/Khomski theory...