1
$\begingroup$

I am trying to fit my model $\omega(\vec{k})$ to the following experimental data to extract three parameters $J_x$, $J_y$, and $J_z$. The data that I am trying to fit 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}}
ldp = 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)", 
     "4T [H,H,0] = [0.45, 0.55], K' -> \[CapitalGamma]"}, 
   PlotLegends -> {Placed[
      BarLegend[Automatic, Automatic, 
       LegendLabel -> "Intensity (a.u.)"], Right]}];

The first, second, and third value of each data point is the x-value, y-value (energy), and intensity respectively. The energy model I am trying to fit is

$$ \omega(\vec{k}) = \sqrt{|S(J_x + J_y)(\cos{k_1} + \cos{k_2} + \cos{(k_1 + k_2)}) +\mu g B_z - 6SJ_z |^2 - |S(J_x - J_y)(e^{-i\frac{2\pi}{3}}\cos{k_1} + e^{i\frac{2\pi}{3}}\cos{k_2} + \cos{(k_1 + k_2)}) |^2} $$ where $S=1/2$, $g=2$, $B_z=4$, and $\mu$ is the Bohr magneton $\mu = 5.788381\times 10^{−2} meV⋅T^{−1}$. To fit this model to the data I convert each x-position to a vector $\vec{k}$ and I can then feed that into my model to compute the energy for that specific vector. The code to do that is

Clear[Jx, Jy, Jz, g, \[Mu], Bz, S]
(* Lattice vectors *)
a1 = {Sqrt[3]/2, 1/2, 0};
a2 = {Sqrt[3]/2, -(1/2), 0};
(* Reciprocal lattice vectors satisfying Subscript[a, i]Subscript[b, \
j] = 2Subscript[\[Pi]\[Delta], ij] *)
b1 = {(2 \[Pi])/Sqrt[3], 2 \[Pi], 0};
b2 = {(2 \[Pi])/Sqrt[3], -2 \[Pi], 0};
(* Bohr magneton μ in units of meV*T^-1 pulled from \
wikipedia*)
μ = 5.7883818012*10^-2;
(* Clean up data *)
data = Select[data, 1.15 > #[[2]] > 0.25 &];
(* Convert x to k *)
x = DeleteDuplicates@*Flatten@data[[All, 1]];
k = Table[(0.5 + x[[i]]) b1 + (x[[i]] - 0.5) b2, {i, 1, Length[x]}];
(* Create and store lists for the y values (energy) and measured intensities *)
y = DeleteDuplicates@*Flatten@data[[All,2]];
intensity = Flatten@data[[All,3];
(* Create a table of the energy spectrum for each momentum vector we created \
above *)
lsw[arr_, Jx_, Jy_, Jz_, S_, g_, Bz_] := 
 Table[\[Sqrt](Abs[
      S (Jx + Jy) (Cos[arr[[i, 1]]] + Cos[arr[[i, 2]]] + 
          Cos[arr[[i, 1]] + arr[[i, 2]]]) - 6 S Jz + g μ Bz ]^2 - 
     Abs[S (Jx - Jy) (E^(-I 2 \[Pi]/3) Cos[arr[[i, 1]]] + 
         E^(I 2 \[Pi]/3) Cos[arr[[i, 2]]] + 
         Cos[arr[[i, 1]] + arr[[i, 2]]])]^2), {i, 1, Length[arr]}]
spectrum = lsw[k, Jx, Jy, Jz, 1/2, 2, 4];

Now I run into a problem. I want to assign a weight to each data point by a Gaussian function $Ie^{-(\omega - y)^2/w}$. Here $I$ is the measured intensity value present in data, $\omega$ is my models energy value, $y$ is the unique energy values in my data set, and $w$ is an unknown width for the Gaussian that I would also like to find via the fit. I am unsure of how to actually proceed from here to properly fit the data like this.

Currently what I am doing is using the intensity values to extract a weighted energy mean and then define a cost function to minimize to extract the three parameters. To check the fit I then take the parameter values I have just extracted and plot the energy spectrum over the intensity plot

weightedEnergy = Mean /@ WeightedData @@@ Transpose /@ Map[Rest, GatherBy[data, First], {2}]
CostFunction[Jx_, Jy_, Jz_] := Sum[Abs[spectrum[[j]] - weightedEnergy[[j]]]^2, {j, 1, Length[weightedEnergy]}]
params = NMinimize[{CostFunction[Jx, Jy, Jz]}, {Jx, Jy, Jz}]
Show[ldp, Graphics[Point[Thread[{x, spectrum /. Last[param]}]]]]

enter image description here

I am wondering if it is possible to do what I mentioned above with weighting the data points via a Gaussian and performing a similar fit procedure as what I have done above to obtain a better fit?

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.