Fitting 3D data with a nonlinear model

Question

I am trying to fit a model to an intensity map to extract three values for the parameters of my model $J_x, J_y, J_z$. Currently, I am running into the issue where I can extract values for these three parameters but they do not provide an accurate fit.

The dataset that I am trying to fit my model to is

data = {{0.01299648, 0.01203211, 0.1263361}, {0.01299648, 0.04910681, 0.0336076}, {0.01299648, 0.09977061, 0.001322289}, {0.01299648, 0.1508783, 0.000499663}, {0.01299648, 0.2008796, 0.000419055}, {0.01299648, 0.2510364, 0.000421737}, {0.01299648, 0.3009251, 0.000178943}, {0.01299648, 0.3508747, 0.0000992}, {0.01299648, 0.3999321, 0.000430162}, {0.01299648, 0.4489179, 0.001252234}, {0.01299648, 0.5002585, 0.000617269}, {0.01299648, 0.5509165, 0.001468457}, {0.01299648, 0.6011173, 0.003723728}, {0.01299648, 0.6498302, 0.004062989}, {0.01299648, 0.6988636, 0.001993023}, {0.01299648, 0.7499531, 0.000721637}, {0.01299648, 0.8010127, 0.000252952}, {0.05334629, 0.01203211, 0.1305249}, {0.05334629, 0.04910681, 0.03503799}, {0.05334629, 0.09977061, 0.001494748}, {0.05334629, 0.1508783, 0.000631434}, {0.05334629, 0.2008796, 0.000516482}, {0.05334629, 0.2510364, 0.000452927}, {0.05334629, 0.3009251, 0.00038714}, {0.05334629, 0.3508747, 0.000419254}, {0.05334629, 0.3999321, 0.000425151}, {0.05334629, 0.4489179, 0.000511058}, {0.05334629, 0.5002585, 0.000683154}, {0.05334629, 0.5509165, 0.001937698}, {0.05334629, 0.6011173, 0.003902016}, {0.05334629, 0.6498302, 0.00309874}, {0.05334629, 0.6988636, 0.001156821}, {0.05334629, 0.7499531, 0.000876003}, {0.05334629, 0.8010127, 0.000494271}, {0.05334629, 0.8507249, 0.000468474}, {0.05334629, 0.9009042, 0.000227273}, {0.09946869, 0.01203211, 0.1314684}, {0.09946869, 0.04910681, 0.03540794}, {0.09946869, 0.09977061, 0.001480958}, {0.09946869, 0.1508783, 0.000518701}, {0.09946869, 0.2008796, 0.00039427}, {0.09946869, 0.2510364, 0.000395253}, {0.09946869, 0.3009251, 0.000357242}, {0.09946869, 0.3508747, 0.000435539}, {0.09946869, 0.3999321, 0.000440639}, {0.09946869, 0.4489179, 0.000453975}, {0.09946869, 0.5002585, 0.000817151}, {0.09946869, 0.5509165, 0.002821699}, {0.09946869, 0.6011173, 0.005799377}, {0.09946869, 0.6498302, 0.003142505}, {0.09946869, 0.6988636, 0.00089875}, {0.09946869, 0.7499531, 0.000935933}, {0.09946869, 0.8010127, 0.000741281}, {0.09946869, 0.8507249, 0.000379727}, {0.09946869, 0.9009042, 0.000452129}, {0.09946869, 0.9499643, 0.000321497}, {0.09946869, 0.9987899, 0.000216047}, {0.09946869, 1.049146, 0.000321083}, {0.09946869, 1.100007, 0.00007}, {0.09946869, 1.151003, 0.000371132}, {0.09946869, 1.20088, 0.001603127}, {0.1513923, 0.01203211, 0.1271927}, {0.1513923, 0.04910681, 0.03466543}, {0.1513923, 0.09977061, 0.001471831}, {0.1513923, 0.1508783, 0.000458085}, {0.1513923, 0.2008796, 0.000323088}, {0.1513923, 0.2510364, 0.000262988}, {0.1513923, 0.3009251, 0.000208331}, {0.1513923, 0.3508747, 0.000261884}, {0.1513923, 0.3999321, 0.000318829}, {0.1513923, 0.4489179, 0.000390393}, {0.1513923, 0.5002585, 0.000796483}, {0.1513923, 0.5509165, 0.003169181}, {0.1513923, 0.6011173, 0.006016299}, {0.1513923, 0.6498302, 0.002961305}, {0.1513923, 0.6988636, 0.00089255}, {0.1513923, 0.7499531, 0.000601942}, {0.1513923, 0.8010127, 0.000622036}, {0.1513923, 0.8507249, 0.00059582}, {0.1513923, 0.9009042, 0.000342461}, {0.1513923, 0.9499643, 0.000365842}, {0.1513923, 0.9987899, 0.000383168}, {0.1513923, 1.049146, 0.000158197}, {0.1513923, 1.100007, 0.0000797}, {0.1513923, 1.151003, 0.000272186}, {0.1513923, 1.20088, 0.000791483}, {0.1513923, 1.24981, 0.000810134}, {0.1513923, 1.298876, 0.001098106}, {0.1513923, 1.35012, 0.001020097}, {0.1513923, 1.400843, 0.00099628}, {0.1513923, 1.45113, 0.001696679}, {0.1513923, 1.487999, 0.00068497}, {0.200413, 0.01203211, 0.1251366}, {0.200413, 0.04910681, 0.03396657}, {0.200413, 0.09977061, 0.001455514}, {0.200413, 0.1508783, 0.000471295}, {0.200413, 0.2008796, 0.000356116}, {0.200413, 0.2510364, 0.000255785}, {0.200413, 0.3009251, 0.000192659}, {0.200413, 0.3508747, 0.000195415}, {0.200413, 0.3999321, 0.000212672}, {0.200413, 0.4489179, 0.000201986}, {0.200413, 0.5002585, 0.000452148}, {0.200413, 0.5509165, 0.002283648}, {0.200413, 0.6011173, 0.005130702}, {0.200413, 0.6498302, 0.002947664}, {0.200413, 0.6988636, 0.000885883}, {0.200413, 0.7499531, 0.000353508}, {0.200413, 0.8010127, 0.000337989}, {0.200413, 0.8507249, 0.000294789}, {0.200413, 0.9009042, 0.000287179}, {0.200413, 0.9499643, 0.000311635}, {0.200413, 0.9987899, 0.000207756}, {0.200413, 1.049146, 0.000158257}, {0.200413, 1.100007, 0.000190184}, {0.200413, 1.151003, 0.000213257}, {0.200413, 1.20088, 0.000336925}, {0.200413, 1.24981, 0.000487695}, {0.200413, 1.298876, 0.000638927}, {0.200413, 1.35012, 0.001054225}, {0.200413, 1.400843, 0.001720866}, {0.200413, 1.45113, 0.00200075}, {0.200413, 1.487999, 0.001241234}, {0.249747, 0.01203211, 0.1205826}, {0.249747, 0.04910681, 0.03260196}, {0.249747, 0.09977061, 0.001261705}, {0.249747, 0.1508783, 0.000333223}, {0.249747, 0.2008796, 0.000305626}, {0.249747, 0.2510364, 0.000248531}, {0.249747, 0.3009251, 0.000227554}, {0.249747, 0.3508747, 0.000242098}, {0.249747, 0.3999321, 0.000213059}, {0.249747, 0.4489179, 0.000171657}, {0.249747, 0.5002585, 0.000269242}, {0.249747, 0.5509165, 0.001302021}, {0.249747, 0.6011173, 0.003488679}, {0.249747, 0.6498302, 0.003051286}, {0.249747, 0.6988636, 0.001461859}, {0.249747, 0.7499531, 0.000595035}, {0.249747, 0.8010127, 0.000388456}, {0.249747, 0.8507249, 0.000291368}, {0.249747, 0.9009042, 0.000241225}, {0.249747, 0.9499643, 0.000237664}, {0.249747, 0.9987899, 0.000151976}, {0.249747, 1.049146, 0.000151201}, {0.249747, 1.100007, 0.000175435}, {0.249747, 1.151003, 0.000146402}, {0.249747, 1.20088, 0.000207344}, {0.249747, 1.24981, 0.000247186}, {0.249747, 1.298876, 0.000406102}, {0.249747, 1.35012, 0.000704678}, {0.249747, 1.400843, 0.000829369}, {0.249747, 1.45113, 0.000937308}, {0.249747, 1.487999, 0.001484896}, {0.2999469, 0.01203211, 0.1134484}, {0.2999469, 0.04910681, 0.03021775}, {0.2999469, 0.09977061, 0.00106177}, {0.2999469, 0.1508783, 0.00030537}, {0.2999469, 0.2008796, 0.000213005}, {0.2999469, 0.2510364, 0.000185747}, {0.2999469, 0.3009251, 0.000164455}, {0.2999469, 0.3508747, 0.000142963}, {0.2999469, 0.3999321, 0.000130967}, {0.2999469, 0.4489179, 0.000167198}, {0.2999469, 0.5002585, 0.000180923}, {0.2999469, 0.5509165, 0.000541689}, {0.2999469, 0.6011173, 0.001467756}, {0.2999469, 0.6498302, 0.00203471}, {0.2999469, 0.6988636, 0.001823599}, {0.2999469, 0.7499531, 0.001339045}, {0.2999469, 0.8010127, 0.0009652}, {0.2999469, 0.8507249, 0.00046882}, {0.2999469, 0.9009042, 0.000292738}, {0.2999469, 0.9499643, 0.000180685}, {0.2999469, 0.9987899, 0.000126662}, {0.2999469, 1.049146, 0.000182039}, {0.2999469, 1.100007, 0.000243545}, {0.2999469, 1.151003, 0.000167506}, {0.2999469, 1.20088, 0.000221046}, {0.2999469, 1.24981, 0.000266948}, {0.2999469, 1.298876, 0.000361939}, {0.2999469, 1.35012, 0.000423017}, {0.2999469, 1.400843, 0.000497565}, {0.2999469, 1.45113, 0.000841964}, {0.2999469, 1.487999, 0.001636413}, {0.351399, 0.01203211, 0.1023983}, {0.351399, 0.04910681, 0.02760775}, {0.351399, 0.09977061, 0.001013449}, {0.351399, 0.1508783, 0.000314347}, {0.351399, 0.2008796, 0.000218643}, {0.351399, 0.2510364, 0.000145264}, {0.351399, 0.3009251, 0.0000988}, {0.351399, 0.3508747, 0.0000886}, {0.351399, 0.3999321, 0.000078}, {0.351399, 0.4489179, 0.000113312}, {0.351399, 0.5002585, 0.0000947}, {0.351399, 0.5509165, 0.000150304}, {0.351399, 0.6011173, 0.000443105}, {0.351399, 0.6498302, 0.000857112}, {0.351399, 0.6988636, 0.001158728}, {0.351399, 0.7499531, 0.001562672}, {0.351399, 0.8010127, 0.001445169}, {0.351399, 0.8507249, 0.000932427}, {0.351399, 0.9009042, 0.00044006}, {0.351399, 0.9499643, 0.000182093}, {0.351399, 0.9987899, 0.000100358}, {0.351399, 1.049146, 0.000133863}, {0.351399, 1.100007, 0.000144129}, {0.351399, 1.151003, 0.00010871}, {0.351399, 1.20088, 0.000123393}, {0.351399, 1.24981, 0.000193871}, {0.351399, 1.298876, 0.000241517}, {0.351399, 1.35012, 0.000296181}, {0.351399, 1.400843, 0.00026523}, {0.351399, 1.45113, 0.000646058}, {0.351399, 1.487999, 0.000963321}, {0.4027665, 0.01203211, 0.09461402}, {0.4027665, 0.04910681, 0.0254652}, {0.4027665, 0.09977061, 0.001031896}, {0.4027665, 0.1508783, 0.000384627}, {0.4027665, 0.2008796, 0.000268799}, {0.4027665, 0.2510364, 0.000208381}, {0.4027665, 0.3009251, 0.000199062}, {0.4027665, 0.3508747, 0.000186287}, {0.4027665, 0.3999321, 0.000159836}, {0.4027665, 0.4489179, 0.000107729}, {0.4027665, 0.5002585, 0.0000923}, {0.4027665, 0.5509165, 0.0000702}, {0.4027665, 0.6011173, 0.000146609}, {0.4027665, 0.6498302, 0.000280341}, {0.4027665, 0.6988636, 0.000459356}, {0.4027665, 0.7499531, 0.000744988}, {0.4027665, 0.8010127, 0.001140008}, {0.4027665, 0.8507249, 0.001200483}, {0.4027665, 0.9009042, 0.000922343}, {0.4027665, 0.9499643, 0.000375711}, {0.4027665, 0.9987899, 0.000140653}, {0.4027665, 1.049146, 0.000098}, {0.4027665, 1.100007, 0.0000772}, {0.4027665, 1.151003, 0.0000815}, {0.4027665, 1.20088, 0.0000899}, {0.4027665, 1.24981, 0.0000992}, {0.4027665, 1.298876, 0.000142484}, {0.4027665, 1.35012, 0.000189181}, {0.4027665, 1.400843, 0.000172183}, {0.4027665, 1.45113, 0.000275446}, {0.4027665, 1.487999, 0.000381214}, {0.4496463, 0.01203211, 0.1094655}, {0.4496463, 0.04910681, 0.03120829}, {0.4496463, 0.09977061, 0.001421187}, {0.4496463, 0.1508783, 0.000589879}, {0.4496463, 0.2008796, 0.000401797}, {0.4496463, 0.2510364, 0.000392448}, {0.4496463, 0.3009251, 0.000374653}, {0.4496463, 0.3508747, 0.000443615}, {0.4496463, 0.3999321, 0.000371195}, {0.4496463, 0.4489179, 0.000203434}, {0.4496463, 0.5002585, 0.000117548}, {0.4496463, 0.5509165, 0.0000764}, {0.4496463, 0.6011173, 0.0000956}, {0.4496463, 0.6498302, 0.000120406}, {0.4496463, 0.6988636, 0.000175619}, {0.4496463, 0.7499531, 0.000325115}, {0.4496463, 0.8010127, 0.0005661}, {0.4496463, 0.8507249, 0.000928296}, {0.4496463, 0.9009042, 0.001128707}, {0.4496463, 0.9499643, 0.000719185}, {0.4496463, 0.9987899, 0.000239313}, {0.4496463, 1.049146, 0.0000817}, {0.4496463, 1.100007, 0.0000617}, {0.4496463, 1.151003, 0.0000532}, {0.4496463, 1.20088, 0.0000802}, {0.4496463, 1.24981, 0.000094}, {0.4496463, 1.298876, 0.00010737}, {0.4496463, 1.35012, 0.0000915}, {0.4496463, 1.400843, 0.00016223}, {0.4496463, 1.45113, 0.000256105}, {0.4496463, 1.487999, 0.000406342}, {0.4967034, 0.01203211, 0.1515964}, {0.4967034, 0.04910681, 0.03684591}, {0.4967034, 0.09977061, 0.001324794}, {0.4967034, 0.1508783, 0.000433766}, {0.4967034, 0.2008796, 0.000275803}, {0.4967034, 0.2510364, 0.000212638}, {0.4967034, 0.3009251, 0.000196184}, {0.4967034, 0.3508747, 0.000205123}, {0.4967034, 0.3999321, 0.000232467}, {0.4967034, 0.4489179, 0.000401981}, {0.4967034, 0.5002585, 0.000176261}, {0.4967034, 0.5509165, 0.0000896}, {0.4967034, 0.6011173, 0.0000941}, {0.4967034, 0.6498302, 0.000100446}, {0.4967034, 0.6988636, 0.000110629}, {0.4967034, 0.7499531, 0.000192725}, {0.4967034, 0.8010127, 0.000359514}, {0.4967034, 0.8507249, 0.000674937}, {0.4967034, 0.9009042, 0.001039717}, {0.4967034, 0.9499643, 0.000770872}, {0.4967034, 0.9987899, 0.000225213}, {0.4967034, 1.049146, 0.0000582}, {0.4967034, 1.100007, 0.0000442}, {0.4967034, 1.151003, 0.0000482}, {0.4967034, 1.20088, 0.0000665}, {0.4967034, 1.24981, 0.0000707}, {0.4967034, 1.298876, 0.0000618}, {0.4967034, 1.35012, 0.0000937}, {0.4967034, 1.400843, 0.000150926}, {0.4967034, 1.45113, 0.000135016}, {0.4967034, 1.487999, 0.000233144}, {0.5504108, 0.01203211, 0.09064252}, {0.5504108, 0.04910681, 0.02602461}, {0.5504108, 0.09977061, 0.000835608}, {0.5504108, 0.1508783, 0.000236157}, {0.5504108, 0.2008796, 0.000142981}, {0.5504108, 0.2510364, 0.000107481}, {0.5504108, 0.3009251, 0.0000878}, {0.5504108, 0.3508747, 0.0000884}, {0.5504108, 0.3999321, 0.000127604}, {0.5504108, 0.4489179, 0.000139343}, {0.5504108, 0.5002585, 0.000122183}, {0.5504108, 0.5509165, 0.0000747}, {0.5504108, 0.6011173, 0.0000848}, {0.5504108, 0.6498302, 0.000123968}, {0.5504108, 0.6988636, 0.000148527}, {0.5504108, 0.7499531, 0.000235699}, {0.5504108, 0.8010127, 0.000372422}, {0.5504108, 0.8507249, 0.000677797}, {0.5504108, 0.9009042, 0.000967788}, {0.5504108, 0.9499643, 0.000557506}, {0.5504108, 0.9987899, 0.000136945}, {0.5504108, 1.049146, 0.0000519}, {0.5504108, 1.100007, 0.0000573}, {0.5504108, 1.151003, 0.0000676}, {0.5504108, 1.20088, 0.0000507}, {0.5504108, 1.24981, 0.0000355}, {0.5504108, 1.298876, 0.0000463}, {0.5504108, 1.35012, 0.000113439}, {0.5504108, 1.400843, 0.000125893}, {0.5504108, 1.45113, 0.000127608}, {0.5504108, 1.487999, 0.000108575}, {0.5999553, 0.01203211, 0.06634204}, {0.5999553, 0.04910681, 0.01829561}, {0.5999553, 0.09977061, 0.000690618}, {0.5999553, 0.1508783, 0.000167709}, {0.5999553, 0.2008796, 0.000126736}, {0.5999553, 0.2510364, 0.0000828}, {0.5999553, 0.3009251, 0.0000879}, {0.5999553, 0.3508747, 0.0000923}, {0.5999553, 0.3999321, 0.0000769}, {0.5999553, 0.4489179, 0.0000776}, {0.5999553, 0.5002585, 0.0000767}, {0.5999553, 0.5509165, 0.0000877}, {0.5999553, 0.6011173, 0.000166282}, {0.5999553, 0.6498302, 0.000232257}, {0.5999553, 0.6988636, 0.000360469}, {0.5999553, 0.7499531, 0.00039869}, {0.5999553, 0.8010127, 0.00053909}, {0.5999553, 0.8507249, 0.000729154}, {0.5999553, 0.9009042, 0.000611591}, {0.5999553, 0.9499643, 0.000244657}, {0.5999553, 0.9987899, 0.0000664}, {0.5999553, 1.049146, 0.0000428}, {0.5999553, 1.100007, 0.0000545}, {0.5999553, 1.151003, 0.0000771}, {0.5999553, 1.20088, 0.0000709}, {0.5999553, 1.24981, 0.000079}, {0.5999553, 1.298876, 0.0000836}, {0.5999553, 1.35012, 0.000146739}, {0.5999553, 1.400843, 0.000180494}, {0.5999553, 1.45113, 0.000171035}, {0.5999553, 1.487999, 0.000283511}, {0.6503139, 0.01203211, 0.08897097}, {0.6503139, 0.04910681, 0.02465827}, {0.6503139, 0.09977061, 0.000902531}, {0.6503139, 0.1508783, 0.00021565}, {0.6503139, 0.2008796, 0.000186667}, {0.6503139, 0.2510364, 0.000151591}, {0.6503139, 0.3009251, 0.000120543}, {0.6503139, 0.3508747, 0.000108839}, {0.6503139, 0.3999321, 0.0000846}, {0.6503139, 0.4489179, 0.000115256}, {0.6503139, 0.5002585, 0.000128018}, {0.6503139, 0.5509165, 0.000132044}, {0.6503139, 0.6011173, 0.000235156}, {0.6503139, 0.6498302, 0.000448416}, {0.6503139, 0.6988636, 0.000524899}, {0.6503139, 0.7499531, 0.000584105}, {0.6503139, 0.8010127, 0.000550121}, {0.6503139, 0.8507249, 0.000462607}, {0.6503139, 0.9009042, 0.000262414}, {0.6503139, 0.9499643, 0.000116691}, {0.6503139, 0.9987899, 0.0000866}, {0.6503139, 1.049146, 0.0000572}, {0.6503139, 1.100007, 0.0000328}, {0.6503139, 1.151003, 0.0000645}, {0.6503139, 1.20088, 0.0000764}, {0.6503139, 1.24981, 0.0000914}, {0.6503139, 1.298876, 0.000131544}, {0.6503139, 1.35012, 0.000152969}, {0.6503139, 1.400843, 0.000160366}, {0.6503139, 1.45113, 0.000227675}, {0.6503139, 1.487999, 0.000410726}, {0.6990286, 0.01203211, 0.1305426}, {0.6990286, 0.04910681, 0.03146357}, {0.6990286, 0.09977061, 0.001037019}, {0.6990286, 0.1508783, 0.000231548}, {0.6990286, 0.2008796, 0.000298593}, {0.6990286, 0.2510364, 0.000231503}, {0.6990286, 0.3009251, 0.000136526}, {0.6990286, 0.3508747, 0.000123731}, {0.6990286, 0.3999321, 0.000157325}, {0.6990286, 0.4489179, 0.000276389}, {0.6990286, 0.5002585, 0.000241361}, {0.6990286, 0.5509165, 0.000154788}, {0.6990286, 0.6011173, 0.000268714}, {0.6990286, 0.6498302, 0.000475129}, {0.6990286, 0.6988636, 0.000607848}, {0.6990286, 0.7499531, 0.000470629}, {0.6990286, 0.8010127, 0.000318165}, {0.6990286, 0.8507249, 0.000204583}, {0.6990286, 0.9009042, 0.000127872}, {0.6990286, 0.9499643, 0.0000904}, {0.6990286, 0.9987899, 0.000180306}, {0.6990286, 1.049146, 0.000121658}, {0.6990286, 1.100007, 0.000130731}, {0.6990286, 1.151003, 0.00015566}, {0.6990286, 1.20088, 0.000097}, {0.6990286, 1.24981, 0.0000895}, {0.6990286, 1.298876, 0.00014314}, {0.6990286, 1.35012, 0.000181547}, {0.6990286, 1.400843, 0.000167488}, {0.6990286, 1.45113, 0.000292798}, {0.6990286, 1.487999, 0.000433997}, {0.7477285, 0.3009251, 0.000053}, {0.7477285, 0.3508747, 0.0000956}, {0.7477285, 0.3999321, 0.000344739}, {0.7477285, 0.4489179, 0.000390532}, {0.7477285, 0.5002585, 0.000392541}, {0.7477285, 0.5509165, 0.000149147}, {0.7477285, 0.6011173, 0.000165453}, {0.7477285, 0.6498302, 0.000452263}, {0.7477285, 0.6988636, 0.000415388}, {0.7477285, 0.7499531, 0.000297813}, {0.7477285, 0.8010127, 0.000194161}, {0.7477285, 0.8507249, 0.000154568}, {0.7477285, 0.9009042, 0.000134434}, {0.7477285, 0.9499643, 0.000110127}, {0.7477285, 0.9987899, 0.000110382}, {0.7477285, 1.049146, 0.000129385}, {0.7477285, 1.100007, 0.000353017}, {0.7477285, 1.151003, 0.000538544}, {0.7477285, 1.20088, 0.00015782}, {0.7477285, 1.24981, 0.0000696}, {0.7477285, 1.298876, 0.000146578}, {0.7477285, 1.35012, 0.000300833}, {0.7477285, 1.400843, 0.000445233}, {0.7477285, 1.45113, 0.000371585}, {0.7477285, 1.487999, 0.00069879}, {0.79343, 0.8010127, 0.000545414}, {0.79343, 0.8507249, 0.000225278}, {0.79343, 0.9009042, 0.000404924}, {0.79343, 0.9499643, 0.000148193}, {0.79343, 0.9987899, 0.000165649}, {0.79343, 1.049146, 0.0000836}, {0.79343, 1.100007, 0.000175279}, {0.79343, 1.151003, 0.000331701}, {0.79343, 1.20088, 0.000197516}, {0.79343, 1.24981, 0.0000473}, {0.79343, 1.298876, 0.000169997}, {0.79343, 1.35012, 0.000461569}, {0.79343, 1.400843, 0.000334411}, {0.79343, 1.45113, 0.000547673}, {0.79343, 1.487999, 0.000890756}, {0.8292615, 1.35012, 0.000118945}, {0.8292615, 1.400843, 0.000102903}, {0.8292615, 1.45113, 0.000157944}, {0.8292615, 1.487999, 0.000248875}}

The first entry in each list is the x-position, the second entry is the energy, and the third entry is the intensity. Plotting this dataset is done via ListDensityPlot

ListDensityPlot[data, 
 ColorFunction -> ColorData["Rainbow"], PlotLegends -> Automatic, 
 PlotRange -> {0, 0.005}, ClippingStyle -> Red, 
 InterpolationOrder -> 0, LabelStyle -> {18, GrayLevel[0]}, 
 FrameLabel -> {"[H+0.5, -H+0.5, 0]", "Energy (meV)", 
   "[H,H,0] = [0.45, 0.55]"}, 
 PlotLegends -> {Placed[
    BarLegend[Automatic, Automatic, 
     LegendLabel -> "Intensity (a.u.)"], Right]}]

The model that I want to fit this dataset to is defined as $$ \sqrt{| S(J_x+\text{Jy}) (\cos (k_1)+\cos (k_2)+\cos (k_1 + k_2))+\mu g B_z - 6J_z S| ^2-\left| S (J_x-J_y) \left(e^{\frac{-2 \pi i}{3} } \cos (k_1)+e^{\frac{2 \pi i}{3}} \cos (k_2)+\cos (k_1 + k_2)\right)\right| ^2} $$ where $S = 1/2$, $g = 2$, $B_z = 4$, and $\mu$ is the Bohr magneton $\mu = 5.788381\times 10^{-2} meV\cdot T^{-1}$.

What I thought would be a quick way to fit this data would be to create a cost function between my model and the energy values swept across then minimize this function to extract the three parameters. So here is how I did that. First I defined how to convert the x-positions into $(k_x, k_y, k_z)$ that my model uses

μ = 5.7883818012*10^-2; (* Bohr Magneton pulled from Wikipedia in units of meV * T^-1 *)
b1 = {2 \[Pi], (2 \[Pi])/Sqrt[3], 0};
b2 = {0, (4 \[Pi])/Sqrt[3], 0};
(* First obtain the measured energy values and the measured x values store them in separate lists *)
ω = DeleteDuplicates@*Flatten@data[[All, 2]];
x = DeleteDuplicates@*Flatten@data[[All, 1]];
(* For the above dataset we are moving along the path K -> M: \
k=x(b1+b2) where 1/3 <= x <= 1/2 *)
k = Table[x[[i]] (b1 + b2), {i, 1, Length[x]}];

Now that I have a list of lists k where the first entry in each list is $k_x$ and the second entry is $k_y$ I can feed these to my model and create a list of functions that depend on $J_x$, $J_y$, and $J_z$

(* Create a table of the spectrum for each momentum vector we created above *)
lsw[Jx_, Jy_, Jz_, S_, g_, Bz_] := Table[\[Sqrt](Abs[
       S (Jx + Jy) (Cos[k[[i, 1]]] + 
           Cos[(-Sqrt[3] k[[i, 1]])/2 + k[[i, 2]]/2] + 
           Cos[(-Sqrt[3] k[[i, 1]])/2 - k[[i, 2]]/2]) - 6 S Jz + 
        g \[Mu] Bz ]^2 - 
      Abs[S (Jx - Jy) (E^(-I 2 \[Pi]/3) Cos[k[[i, 1]]] + 
          E^(I 2 \[Pi]/3) Cos[(-Sqrt[3] k[[i, 1]])/2 + k[[i, 2]]/2] + 
          Cos[(-Sqrt[3] k[[i, 1]])/2 - k[[i, 2]]/2])]^2), {i, 1, 
    Length[k]}];
    spectrum = lsw[Jx, Jy, Jz, 1/2, 2, 4];

Now I define my cost function and minimize it to extract the parameter values

CostFunction[Jx_, Jy_, Jz_] := 
 Sum[Sum[Abs[spectrum[[j]] - ω[[l]]]^2, {l, 1, 
    Length[ω]}], {j, 1, Length[energies]}]
(* Minimize this cost function to attempt to extract the parameters \
Jx, Jy, Jz, and g. Ideally, we know what the parameters should be but \
in this case, I only know that g = 2. These parameters will be \
extracted in units of meV.*)
params = NMinimize[{CostFunction[Jx, Jy, Jz]}, {Jx, Jy, Jz}]

This gives me the output for the three parameters

{110.945, {Jx -> -0.000561511, Jy -> 0.000561303, Jz -> -0.0957132}}

Looking at these values I can tell they are not going to be a good fit due to the very small values for $Jx$ and $Jy$ and I am correct with my assumptions when I plot the lsw function as a white line over the intensity plot with these values. I was hoping someone would be able to help diagnose where I am going wrong with this fitting procedure? I have also tried using the function NonlinearModelFit and have only gotten errors where it returns a list of complex-valued energies which I do not want. Any help is appreciated!

Answer 1

It wasn't immediately obvious to me from your post what you were trying to achieving (see comments above), but assuming you want to fit e(k) based on some experimental values for intensity - this might help you get started.

First, there's only a handful of kpts in your dataset so let's make the task somewhat easier by cleaning some outliers:

data = Select[data, 1.15 > #[[2]] > 0.25 &];

Second, you need to somehow pass the intensity information - right now your code is just minimizing the difference b/w the energy bin values (which are the same across), and thus returns a flat line at the average energy.

Here, I used the intensity value to extract a weighted energy mean - presumably this will be easy to extend to a weighted LeastSquares or similar if you prefer.

weightedEnergy = 
 Mean /@ WeightedData @@@ 
   Transpose /@ Map[Rest, GatherBy[data, First], {2}]

This means we'll use a slightly simpler CostFunction:

CostFunction[Jx_, Jy_, Jz_] := 
 Sum[Abs[spectrum[[j]] - weightedEnergy[[j]]]^2, {j, 1, 
   Length[weightedEnergy]}]
params = NMinimize[{CostFunction[Jx, Jy, Jz]}, {Jx, Jy, Jz}]

This returns a sensible fit:

Show[ldp, Graphics[Point[Thread[{x, spectrum /. Last[params]}]]]]