Antworkingheatmap.py

#%% loading functions
import numpy as np
from scipy.constants import h, c, e, m_e
import matplotlib.pyplot as plt
from scipy.integrate import quad_vec


# Calculation of absorptive scattering factor

# Accelerating voltage, volts
V = 200000.0
# lorentz factor
gamma = 1 + (e*V)/(m_e*c**2)
# electron velocity
v = c*np.sqrt(1 - 1/gamma**2)

Bvalues = np.array([0.1, 0.15, 0.2, 0.25, 0.3, 0.4, 0.5, 0.7, 1, 1.5, 2, 2.75, 4])

parameterArray = np.array(
[[[2.05727084e-05,5.06271546,2.92660922e-05,0.930335119,0.0135216077,0.116889587,-0.0134867487,0.11688959,-4.28449381e-06],
[2.6592243283279636e-05,5.675920610536081,4.130117211737221e-05,1.1561428379491518,0.013237729128333478,0.14711028397873144,-0.013186438026856262,0.14710018609168965,-8.890601389432829e-06],
[3.094433693645993e-05,6.234324817266906,5.227226160566411e-05,1.3783303633242647,0.013203802504045355,0.18078209428506783,-0.013136213215876366,0.18076696247311716,-1.34649139404092e-05],
[3.472221638375163e-05,6.67192870487392,6.276013026315342e-05,1.5609827629691562,0.01366081200028202,0.21669903574797514,-0.013576928091629515,0.21673628091617256,-1.907410734308862e-05],
[3.7387039310128706e-05,7.108955426633964,7.238477024831784e-05,1.7525657321590964,0.014266764691914282,0.23995081001354993,-0.014166705634761956,0.23994325994101792,-2.4227909497657573e-05],
[5.686276722530545e-05,4.217179147253889,0.00014121749863037664,0.1628081928157759,8.215506861413351e-05,1.3007355199578656,1.724842592333466e-05,11.294375249570532,-6.900475814885715e-05],
[6.372670353980777e-05,4.75858514106865,0.00017974342790246287,0.18676055402906203,9.934323722441242e-05,1.5236084830016605,1.7041819697544292e-05,12.166303181676039,-9.262357149361149e-05],
[7.504571935584617e-05,5.612240810401235,0.0002693018320400263,0.209405177556215,0.00013245033374742422,1.8742699895183625,1.631303558069684e-05,13.556284171063474,-0.00015711458113931156],
[5.2951255424306816e-05,10.400319968670631,0.0001463397601958656,0.4446003956148727,0.00019711387648741269,0.43043595254610234,0.0001791245310277009,3.312162062819229,-0.0001517811250961153],
[0.00011353735083479875,8.922102228860423,-0.023133231625010003,1.0241193158910242,0.01156857083770998,0.8203087817301588,0.012481414115880612,1.2024470855975666,-0.0004868655294797641],
[0.00038220319363342197,0.4115891061827376,0.0004970315992617557,0.4123823037162079,0.0002888114791738547,4.369477262148248,5.600760769086176e-05,12.655079503071542,-0.0005823960126630608],
[-0.0006916534680506142,0.5308951576865517,0.002079257256469036,0.4570150315706604,0.0003400770190688078,5.287705956775111,4.1146510735290827e-05,15.140920933098775,-0.001006571130385495],
[-0.05131362507630703,1.5329332979865766,0.00015842659272586148,10.655895238766021,0.02846606978149747,1.222594337572545,0.025380020048239398,1.8168698586946666,-0.0017729751048727567]],

[[4.7829055708734086e-05,2.7595555850655273,9.788437985389019e-05,0.6486179268169794,-0.0901274219618626,0.11652558501083393,0.09025727527187816,0.11648809426798096,-2.3601917492010693e-05],
[5.833219120985202e-05,3.1018161111742186,0.0001363572486344445,0.7823025032382582,-0.01291331122903196,0.13863588711632333,0.013107197839507888,0.13817834839112347,-5.091509851603294e-05],
[6.498731243312285e-05,3.390742873636509,0.00017042609787800607,0.905190359722796,-0.016924646553910922,0.17043718590764692,0.0171791330721187,0.16975650974791381,-7.717878841964527e-05],
[6.80699916180378e-05,3.6825555132552092,0.00020204865726268774,1.0250611471180817,-0.019472183268383847,0.20702059713282323,0.01978415031664284,0.20596799377663202,-0.00010193853921538202],
[7.11576274996396e-05,3.9079909020373873,0.00023117087067651293,1.1192537501605273,-0.020057850882434333,0.22490988623202243,0.020434303948419757,0.22353628440912293,-0.0001371324252820972],
[0.00010808661494121703,3.1688732668055613,0.0006209008729659266,0.11411686270664517,0.00027616705346043856,1.0401129999839986,9.060768183041247e-06,8.313705689690527,-0.00036300089389837127],
[0.00010901473397391424,3.6343810869317537,0.0008307552392888151,0.11963432336583554,0.0003265621746658818,1.1962250401170804,5.397681796624581e-06,10.203058639176907,-0.000524737094629172],
[0.00010923782344592806,4.28157961324478,0.0014742121739780144,0.10545767835528559,0.000416720054657845,1.4161628094201133,2.5567409980496686e-06,13.326990560965184,-0.0010930612205451945],
[1.0503955019848938e-05,10.383269515654368,0.010047024241472101,0.42479803538493244,-0.008717944127776402,0.43524521324582033,0.000378269604644287,2.8291704245104907,-0.0006127030568226593],
[3.935912979094519e-05,8.827152832931494,-0.05982182850222017,1.1632374937104037,0.013876270916160146,0.8007425367477211,0.04842651843747552,1.2471640030887814,-0.0011671358147643189],
[-0.9103008388211893,0.4115478777219137,0.9145972175208514,0.4103010878135971,0.0006960668430528053,2.8250610426494642,1.3195717368136461e-05,12.872382445386041,-0.0034643848716509763],
[-0.0022461408141845576,0.6148398221702139,0.007569207870322816,0.3938723548655265,0.00040563901526827326,5.065991262469326,-2.2848216400857325e-05,14.945679514622565,-0.003950001374797637],
[-0.12429830995803533,1.5367072763109033,-7.425568561900528e-06,10.071094342801926,0.07055039141858124,1.1928511324151654,0.060931434304443574,1.8239355376130337,-0.005165253596623765]],

[[6.899316716358343e-05,10.055910023812487,0.00020140788793549379,0.9504553889808925,0.016874985622478477,0.12059118133008473,-0.016563094195507513,0.12059121674112322,-3.0407987157385666e-05],
[9.139293706650853e-05,12.158191009269101,0.00026948890712517236,1.1359847389063966,-0.027992114673802235,0.14340159437931524,0.028448762088156727,0.1434016616728456,-7.676496204725523e-05],
[0.00031998347445817677,0.8934772748948614,0.0005876656054360649,0.12453444272421371,6.441833552057349e-05,26.917210338224923,0.00012154823140557894,4.018484126663412,-0.00018317775577433368],
[0.00012388900147985264,16.361707747612606,0.0015208407739029548,0.20269284338189913,0.0003546629186066826,1.5784336550878908,-0.0007873070226365748,0.20270068445044562,-0.0001536092088114624],
[0.0001421879177629622,5.301691562965971,0.000881753451766597,0.15645866291465024,0.0004120655044981547,1.1627933761263145,8.494835881093389e-05,29.997455489689724,-0.0003212365639809233],
[0.00014086954517020361,7.994096363011822,0.0011281031788319743,0.22895302789366226,0.0004459917638697011,1.6588959433022668,9.115405739662055e-05,34.901837422756834,-0.0003567637266337946],
[0.00015121775785576332,10.83459034678902,0.001375342180923692,0.28451595278617847,0.0004753853051587621,2.1186508957048815,9.220605733309762e-05,39.29797932421858,-0.00042280602163096125],
[0.00018972537431715173,15.057475604172506,0.0018652626012271123,0.35248561344786317,0.0005351859385604709,2.8312726638813612,9.310655595081231e-05,45.511529299395214,-0.0006255428180848224],
[0.0002833680498736884,29.84326486365283,0.0015933265919722728,0.48303574589909654,0.0008961159704331931,0.4829872026957848,0.0005874690279222691,4.572783741630541,-0.0008196640220787028],
[0.0003449072272077114,34.617961260313116,0.00214210298467308,0.5945058128871764,0.0006789414412269691,6.603967842264421,0.0013493459172302922,0.5944936432363365,-0.0013190439966178473],
[0.00237513793585757,0.5224433741039627,0.0008382979426078352,6.959797605451152,0.002663649357134064,0.5223728888896547,0.00044859796547806394,34.78401394658654,-0.0025909917506728116],
[0.0025880982619846636,0.8903166932334492,0.0009392191079126414,13.705578408267584,0.0027943683405321144,0.8904347423943366,0.0003107726292354461,50.298643296304355,-0.0022213704700770613],
[0.004257806287814436,0.7981051563605709,0.004219827182499092,0.7981215175859627,0.0012311382408488724,15.366884349140156,0.00040257791548447495,51.029855563905116,-0.004793228743908]],

[[0.00017326947693371072,8.739744183745835,0.0003167617746883571,0.9054672390071232,0.016902460151066428,0.11465129988207035,-0.01636080945425545,0.11465135896436963,-5.73672991315459e-05],
[0.00024091177656421724,9.605964275874218,0.00042694645280045944,1.0674847691140947,0.04412830861039953,0.13163366368351423,-0.043336940233553556,0.13163368351207544,-0.00014953867592954476],
[0.0005011371627911596,0.8456512276257114,0.0010301017473829397,0.11482528166311266,0.0001738901891613387,16.48055190967877,0.0002487484372106858,3.986512681790138,-0.000342563946456167],
[0.00034325022823151216,11.425738693852027,0.0018209135118873745,0.18335101059060985,0.0005707200086021362,1.5098566777783033,-0.0005602391559458285,0.18333652561961888,-0.00029608276996779143],
[0.00032979488867662606,5.246695740886023,0.0015289912108780689,0.1472978706830542,0.0006354278092113368,1.1355641489462232,0.00021360326957887914,18.39072169306789,-0.0005752005098716375],
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[[0.0003087990074547848,6.577166995466867,0.00046365416970969804,0.8551669993236387,0.49117676823879025,0.10755719463315302,-0.49035299630045537,0.10755720214340239,-9.906156120228823e-05],
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[[0.000599830247,4.09083533,-0.0516899155,0.759840549,0.00154067734,0.0950248602,0.0525594258,0.759840441,-0.000236246113],
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[0.011280205348902278,7.79385074335794,2.0394355951691256,0.019509971913836625,0.2192860220919225,0.5611387134119755,0.07597379961253312,2.197736805293349,-1.7496409424839392],
[0.009225635793489425,9.77930925153045,5.88726870779917,0.010000000003480282,0.2530382521571995,0.6920343229902761,0.08077867528127855,2.653717762416274,-5.53781675380168],
[0.006378891814573047,12.988848886985085,8.472360183990295,0.010000000000000002,0.2725324030742906,0.8716977462975415,0.07627475343747236,3.2646869681584323,-8.053900930697875],
[0.5127336417043787,5.688148259686638,9.999999995024096,0.014872686980655548,0.3039005846227824,1.3297050497364569,-0.4622772143591601,5.688135109877657,-9.448305895862358],
[0.2285634706957127,5.06539771713524,3.1908186246759245,2.902676020631462,9.999999827373442,0.026731303438846842,-3.08829269801775,3.0978293051072145,-9.27217177993377],
[0.04644532777522883,8.532572231955516,0.9558984128083723,1.434094123451581,9.999999179407672,0.10858861033509051,-2.5921322457295597,0.5097622954537557,-7.16529742620022],
[0.02829930942186389,12.846821801180699,1.964873063754302,1.5714593479166372,9.99999933206589,0.17489918851402075,-3.8548371605035636,0.854921586652805,-6.756083867045508],
[0.012515110384920631,19.999999965878366,2.762980187350294,2.0449468592848246,9.999999999999808,0.2878607961286217,-4.999999203368255,1.2857495377081738,-6.239434115425432],
[1.766455213810918,3.0478851917737897,5.295051100329305,0.3203046346950148,9.653525306247495,0.32030881748210055,-4.999999953240959,1.4596718972226903,-9.999988908068632]],

[[0.041276136458104636,1.3657692330314646,0.009946226772112194,4.844244613060402,0.26993858407394317,0.04482305692998145,0.103678968888608,0.34865978529098823,-0.10358302165660643],
[0.15021013341959338,0.357325202936783,0.05883132595810082,1.4309017287273436,0.5569915045909946,0.028306959204395386,0.013570889634341311,5.056886466723485,-0.36898312392532584],
[0.047602196787452186,2.9953992772140743,0.5724953838966115,0.05773608772525161,0.6385780617878254,0.621745875549985,-0.47661379395043174,0.6217466967181904,-0.2974672286397264],
[0.07334902208786932,1.9396729279688754,1.3710210242325882,0.02294352623377606,0.012071161871915205,6.7787965545961635,0.20209378472684705,0.4939125725714647,-1.109430941765449],
[0.011107591206644363,7.655045911684653,2.155468361620799,0.01864566915036447,0.2228800693506715,0.5583479764951805,0.0779510831916209,2.1651527149539187,-1.8615295429451215],
[0.008819320836336247,9.728810315489545,5.988099883225587,0.010000000003979475,0.2568677320785647,0.6939352492233317,0.0818738060722684,2.6290075000240645,-5.632233484766775],
[0.005987005405046692,13.03796341960339,8.613039789532227,0.010000000000000002,0.2774032190465192,0.8745963996851909,0.07647206313033335,3.2382061180163246,-8.187188543651969],
[0.510575689787312,5.638410041811966,9.99999999996217,0.015118348616588622,0.30977377654175714,1.3296135908279627,-0.4613653784559859,5.638444811389392,-9.439080503498523],
[0.6573723652396192,4.379251910860685,2.881791928745599,2.9919219128787176,9.999999974458992,0.027223801273477184,-3.2055513620856435,3.3240573159139517,-9.258987459343636],
[0.044080837511100514,8.547047701656023,0.9994021518735564,1.418629627656645,9.999998749848476,0.10960460290271225,-2.6140901942529364,0.5214621903219517,-7.166784422231077],
[0.02570821275116613,13.137824092370314,1.9884280008204482,1.5733280098485447,9.999999877148193,0.17652162414889777,-3.8656789356791084,0.8629106313390866,-6.747337052902584],
[0.010896926394938226,19.999966431614418,2.7610954567912094,2.0510048015178515,9.999998778782537,0.2913624229875725,-4.999998379872725,1.2935712353501148,-6.216108276762496],
[1.7719599099360004,3.0386307074420666,5.302918152114842,0.32244863799592904,9.660893082653399,0.3224348428709812,-4.999999822734273,1.4699240453111766,-9.999990560727664]],

[[0.04248508776035533,1.3474503226962606,0.009985236165992945,4.708713956611822,0.27578844420883275,0.04437762798350099,0.10508683136487955,0.34588778233573625,-0.10687940067301215],
[0.15232845172968337,0.35438984666562356,0.060567705477608334,1.4108175165294514,0.5738970882908974,0.02775307132509095,0.013587631022940527,4.917420500659587,-0.3832086587811783],
[0.048613902357422904,2.913510486951606,0.5870151998988774,0.056790198947934474,0.6399791262517164,0.6172847879642038,-0.4752106730907248,0.6172802243813622,-0.3081579251261842],
[0.07539507047695881,1.9068700366666995,1.4380959738246823,0.02206880954579631,0.011863435913882878,6.6199016241836315,0.20526244441724661,0.4906214068633422,-1.1729635066304975],
[0.01082550145266017,7.494694217753986,2.298638578928642,0.017634106821138685,0.22663066966886725,0.5549727791876753,0.07993330527406613,2.127085093288371,-2.000811478398264],
[0.008262791864584656,9.69164412745251,6.089479626551656,0.010000000062115485,0.2608504072290415,0.6960212702891198,0.08266291737729624,2.601045683527891,-5.72717649563642],
[0.005470331825412543,13.161294320969471,8.75416109374214,0.010000000000000002,0.2825357101756364,0.8775691348161221,0.07616519126858903,3.2091948292433883,-8.320910812690299],
[0.5114737333154815,5.580526677136968,9.999999999670932,0.015364097622726601,0.315711272354334,1.3280724720507904,-0.4640696247994276,5.580524131168117,-9.429880016721613],
[1.1053312058412466,4.150010337111742,2.5997390855509726,2.9841501019706578,9.999999974389151,0.02769508823639962,-3.3691265773818166,3.4164837752721593,-9.246358942826223],
[0.04087124804419958,8.598965372846266,1.048103598385994,1.4007626976574032,9.999998897714937,0.11057826122514801,-2.640208539268242,0.5340165612956278,-7.169623402976645],
[0.023397417817392196,13.423135888123582,2.4114508485978177,1.499089810771106,9.999999465579359,0.18929947410771664,-4.496369410876912,0.8773253731780509,-6.520014901854577],
[0.009045190283582466,19.970185906667364,2.7769982824861232,2.0523327410396814,9.999999237029233,0.2933197110445819,-4.999997797412416,1.3050067953784468,-6.212386938309094],
[1.762897037730318,3.028105070527188,5.31942874734719,0.32460997468621994,9.671322135832144,0.3246127612635495,-4.99999589740562,1.4752255946795056,-9.999867673430888]],

[[0.04351826190993843,1.3683044058323683,0.00943535918925626,4.935064807891954,0.2800839521463142,0.04502601938017318,0.10595214393748682,0.34978235306877126,-0.10700290359123557],
[0.1536030592517035,0.3587211123941383,0.06199076919139458,1.4321162885786225,0.5755269902970996,0.028546710058083118,0.01280307392731573,5.171557342901493,-0.3797323791809116],
[0.04848154318377275,2.938735987322879,0.5954108155843183,0.05727950568515997,0.6413209004021725,0.6249134904132755,-0.47410929395538726,0.6249166276215479,-0.3106314519407148],
[0.07629409877032246,1.9308797931906738,1.4004060332042603,0.02339099485790314,0.01112548021179475,7.092893831611434,0.20708708017342423,0.4991071398342221,-1.1279217592332325],
[0.010244253384964251,8.059854809393565,2.1849868467639197,0.01913294400714995,0.22894470670180458,0.5650471716073351,0.08037230014256473,2.1517949325440546,-1.8790447108187827],
[0.008348034030293794,10.254412009091709,6.1926545579023875,0.010000000023234597,0.265747532808071,0.700507777654128,0.08304335201528452,2.6021225377159403,-5.823799668434224],
[0.005906819615124677,13.663467825451281,8.894786881583759,0.010000000001179373,0.2886422401940182,0.8819445913537013,0.07559873982753605,3.215313231690215,-8.454227055531412],
[0.5106157302722465,5.826787382596295,9.999999999636497,0.015623230457975637,0.32313489587770355,1.3325680287003137,-0.4646486516373664,5.826802698184146,-9.420385367814037],
[1.0968372779898294,4.212888840708117,2.6062613731230377,2.9981775419925567,9.999999838278255,0.028199461265832445,-3.3622515372743393,3.4448768235408473,-9.23325727121474],
[0.040868884618113714,9.04901015370629,1.0774243324295756,1.3976234212770018,9.999999039669312,0.11154933900929913,-2.646961114194985,0.5434779817942638,-7.171081776158361],
[0.024937569348151667,13.899350242682418,2.5370700635669765,1.4902769660070359,9.999997927259367,0.1933397071480669,-4.64982234015035,0.8887737583535769,-6.470319308966196],
[0.01109379967841034,19.96052670738855,2.794295037455628,2.068875367643252,9.99999987150414,0.29603522110751235,-4.999998121649098,1.3215267873659566,-6.2057203487393995],
[1.7499534737213025,3.089946845675583,5.350555024294258,0.3286905585923409,9.68213423947822,0.3286966896786307,-4.99999975646172,1.490739329958535,-9.99987338029304]]]
)

# The lobato parameterisation of the the elastic scattering factors is used, here stored in a text file
lobato_array = np.array([[6.473848488352918e-03,-4.901925767802290e-01,5.732841603908765e-01,-3.794033014839905e-01,5.544264747740791e-01,2.785198853791489e+00,2.776204283306448e+00,2.775385910506251e+00,2.767593028672588e+00,2.765118976429275e+00],
[3.057451160998355e+00,-6.200447791273253e+01,6.400555370846145e+01,-5.001325785427806e+00,1.517988287005264e-01,1.089672487260788e+00,9.398387981431211e-01,9.252890343862655e-01,8.229474987086506e-01,5.773931106754022e-01],
[3.926222728861479e+00,-4.548619626399980e+00,2.193353128786585e+00,6.994512650339657e-02,2.098642248519376e-03,8.142760135172804e+00,4.989410770078558e+00,4.144289992394109e+00,4.019223150656802e-01,1.564790347198236e-01],
[3.398249705570541e+00,-1.908668860956967e+00,3.907021175392274e-02,-1.116310102107145e-02,9.462044653575231e-03,4.442701786224095e+00,3.324515425264230e+00,1.897728803482149e-01,8.719186146446036e-02,8.278090600413406e-02],
[1.472792486393293e+00,-4.019330421993871e-01,3.059989569826894e-01,1.961442171731680e-02,9.771771060882025e-04,3.749740482818191e+00,5.880665361396738e-01,5.156396131030110e-01,1.213775700806037e-01,6.809824121603139e-02],
[1.244660886213433e+02,-2.203528570789638e+02,1.952353522804791e+02,-9.810793612697997e+01,1.420230412136232e-02,2.421208492560056e+00,2.305379437524258e+00,2.048519321065642e+00,1.933525529175474e+00,7.689768184783397e-02],
[5.813271507025561e+01,-1.475424090878127e+02,1.301430656496395e+02,-3.961956740841543e+01,1.059577633314809e-02,1.700448564134711e+00,1.559038526017404e+00,1.415768274731469e+00,1.278418182054558e+00,5.655877984748055e-02],
[2.994740452423624e+01,-7.761012662552783e+01,9.988177646231442e+01,-5.121270055056731e+01,8.196189544460320e-03,1.302839878800107e+00,1.157941052583095e+00,1.009885493380251e+00,9.433279714332660e-01,4.331976113218256e-02],
[9.489848945035248e-01,-3.013339230435549e+01,5.279650781273386e+01,-2.270627037952724e+01,6.569976645314410e-03,1.458829331986459e+00,6.887799931876800e-01,6.542398693466957e-01,6.148361308119943e-01,3.428374194950112e-02],
[5.827411922209074e-01,3.706765618410549e-01,-5.467449673508092e-01,4.140526824802081e-01,5.199030808639931e-03,1.281185731438772e+00,4.445208971704776e-01,1.986508755104810e-01,1.854772466562765e-01,2.757383820338858e-02],
[2.367006039467926e+01,-2.185317861597425e+01,5.924994481089464e-01,-2.446522903102440e-02,4.839502217065358e-03,8.451487735146031e+00,8.040966004742982e+00,6.249960005263150e-01,1.324503949472964e-01,2.339943620498786e-02],
[4.855010476871495e+00,-2.662209064768437e+00,4.780012360851080e-01,-7.023070646920648e-02,3.989058281040198e-03,5.946392738424565e+00,4.171303125206979e+00,3.982698081503743e-01,1.618861858374742e-01,1.953450563631035e-02],
[2.834095616075075e+00,-4.280041333782610e+00,4.421916805483114e+00,-3.457744718964006e-02,3.523859414060799e-03,6.662350239805332e+00,5.512947222240214e-01,5.093289634459738e-01,1.117848374253312e-01,1.676023518052574e-02],
[2.871891426116124e+00,-2.061735011951735e+00,2.171140242044787e+00,-6.630736330588019e-02,3.010707096705137e-03,5.084871036429896e+00,4.291781853051262e-01,3.664854341921622e-01,1.197106112969034e-01,1.439945361283975e-02],
[2.791518400231514e+00,-4.365068378238221e+00,4.435584555166990e+00,-8.096357733994739e-02,2.679000179664404e-03,3.900659618654661e+00,3.298259683771500e-01,3.060899565058886e-01,1.080832325459728e-01,1.258944953311868e-02],
[2.679714156101995e+00,-4.742528222307552e-01,5.148359489896877e-01,-9.583600249907225e-02,2.488719638189352e-03,3.068891211999711e+00,3.782167021858090e-01,1.887218119025488e-01,9.233705900314941e-02,1.119208772411441e-02],
[2.566248399800203e+00,-3.388763508285917e-01,1.145845587555150e+00,-9.231093165470796e-01,2.291680020410422e-03,2.415949203656124e+00,4.214142393102160e-01,1.095924049758303e-01,9.909554582267530e-02,9.996659489275210e-03],
[2.459817464140685e+00,-3.641981770769951e-01,2.505844772224745e-01,-5.774370295443458e-02,2.301438668285837e-03,1.940046319888566e+00,3.992410678843988e-01,1.174724062744122e-01,5.678037260236218e-02,9.155798329207080e-03],
[5.811078786014548e+00,-5.025370965394226e+01,4.886094120598425e+01,7.406285920483829e-02,7.278027386262219e-04,1.266914833990368e+01,3.956410396981665e+00,3.683850595771545e+00,1.074585175695628e-01,6.655767893915011e-03],
[2.117811615241595e+01,-3.390438243174689e+02,3.227569585232967e+02,6.500776738969967e-02,6.558743665785935e-04,6.396086194317362e+00,3.740247138917495e+00,3.648884499226053e+00,9.450906345146735e-02,5.985206198837586e-03],
[1.260351865721486e+01,-2.768753820537026e+02,2.688716039073428e+02,5.568241788974580e-02,5.770712551453996e-04,6.156256153638529e+00,3.088735542666793e+00,3.027276632985483e+00,8.188747483752157e-02,5.382898320507976e-03],
[8.575957752381299e+00,-2.103315634653049e+02,2.060971726015414e+02,4.777739489772619e-02,5.057164844803206e-04,6.007806688755818e+00,2.602858567452130e+00,2.553523450511056e+00,7.114294840241539e-02,4.856284393837261e-03],
[6.527684332347890e+00,-2.004305768291727e+02,1.980150538899940e+02,4.139115180640056e-02,4.474550243298565e-04,5.835524793524481e+00,2.232559523811276e+00,2.197860185942290e+00,6.239738768285473e-02,4.403836490799580e-03],
[3.028317848436913e+00,-9.553939330814323e+01,9.617615623521981e+01,3.597773159579877e-02,3.914928907184223e-04,8.359115043146318e+00,1.802637902641032e+00,1.775094889570472e+00,5.481444121431137e-02,3.998289689160348e-03],
[4.374175506331227e+00,-1.609255109187795e+02,1.602733080603223e+02,3.123038610491624e-02,3.469660210209380e-04,5.510317055049282e+00,1.687982164024339e+00,1.666140477777024e+00,4.833703903212770e-02,3.647469600530682e-03],
[3.798100908368596e+00,-9.168935493816872e+01,9.144542521554298e+01,2.727543440276047e-02,3.033798543837710e-04,5.317126458994331e+00,1.497130948847481e+00,1.468092418044105e+00,4.272478501289549e-02,3.327918552318741e-03],
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[6.434013247972842e+00,2.970999705357888e+00,4.607966517878480e-01,7.240331257754553e-03,6.362366643091279e-08,1.025968513072979e+01,1.131774523061633e+00,1.127759353623823e-01,5.274595833531321e-03,6.192638240887597e-05],
[6.210708267840199e+00,3.039344538256235e+00,4.373399844391628e-01,6.807147641473166e-03,6.182293277428513e-08,1.002141058591628e+01,1.103998801558834e+00,1.072189731624539e-01,5.024321308506824e-03,6.005069941971958e-05],
[6.004315983089865e+00,3.094316545314180e+00,4.128084877164170e-01,6.408916578800669e-03,6.730073762160754e-08,9.817930467191061e+00,1.069787050680294e+00,1.016352914486458e-01,4.795248468887250e-03,6.028065987264629e-05],
[5.200617168103042e+00,3.498494403671440e+00,4.054149311313203e-01,6.223437008265299e-03,6.008593295434409e-08,1.120383101944067e+01,1.128463695469282e+00,9.893334726861316e-02,4.659174319779507e-03,5.725906389623341e-05],
[5.025338600360291e+00,3.518439882851541e+00,3.819503494462722e-01,5.821102080962110e-03,6.526093388297119e-08,1.097721906976287e+01,1.084772148326308e+00,9.367468748538359e-02,4.426823468734856e-03,5.738568370505115e-05],
[5.346561002606197e+00,3.224684665609108e+00,3.494617526254452e-01,5.292517567488151e-03,6.159176302362374e-08,9.231183797524494e+00,9.728362291938352e-01,8.705962209665623e-02,4.130119644650323e-03,5.494638944496757e-05],
[5.225823503340047e+00,3.228188739952953e+00,3.270988788238514e-01,4.888825755922390e-03,5.212722694193633e-08,9.071357066187183e+00,9.311242774566348e-01,8.207206020598455e-02,3.890296559934166e-03,5.103679766366416e-05],
[4.586412477535479e+00,3.518695956651012e+00,3.192617142012054e-01,4.729796479856258e-03,5.513244808062041e-08,1.031861020923352e+01,9.572788378292328e-01,7.968074314438807e-02,3.771931999968489e-03,5.097508603760111e-05],
[4.457994806754126e+00,3.508672126376623e+00,3.013753805283154e-01,4.407637462144819e-03,4.887879032285916e-08,1.008906933834013e+01,9.194464473616226e-01,7.564191777863293e-02,3.570264651248093e-03,4.804951743958113e-05],
[4.338975764011709e+00,3.491850988964034e+00,2.814710990162868e-01,4.002092714990461e-03,5.529298081955140e-08,9.963309372238362e+00,8.780839569829744e-01,7.116371157687776e-02,3.321524042150777e-03,4.851031794603728e-05],
[4.227294704031012e+00,3.472492275106616e+00,2.648222294968986e-01,3.690728778331929e-03,6.258714262002473e-08,9.724006020400314e+00,8.428737759602104e-01,6.735347439748511e-02,3.123646062633208e-03,4.912170970176192e-05],
[4.109517024430204e+00,3.457991325227507e+00,2.470873512223867e-01,3.304239209409952e-03,5.991049262319316e-08,9.677359945101202e+00,8.069400424708172e-01,6.328154369541671e-02,2.875446396412092e-03,4.706791536975981e-05]])

# %% subroutines


# scattering factor f, Lobato parameterisation
def lobato(s, Z):
    ab = lobato_array[Z-1]
    g = 2*s    
    f = ab[0]*(2 + ab[5]*g**2)/(1 + ab[5]*g**2)**2 + \
        ab[1]*(2 + ab[6]*g**2)/(1 + ab[6]*g**2)**2 + \
        ab[2]*(2 + ab[7]*g**2)/(1 + ab[7]*g**2)**2 + \
        ab[3]*(2 + ab[8]*g**2)/(1 + ab[8]*g**2)**2 + \
        ab[4]*(2 + ab[9]*g**2)/(1 + ab[9]*g**2)**2
    return f

# Bird & King integrand
def integrand(sx, sy, s, M, Z):
    s1 = np.sqrt((s/2 + sx)**2 + sy**2)
    s2 = np.sqrt((s/2 - sx)**2 + sy**2)
    s_square = sx**2 + sy**2 - (s**2/4)
    result = lobato(s1, Z)*lobato(s2, Z)*(1 - np.exp(-2*M*s_square))
    return result

# double integral part 1
def integral1(sy, s, M, Z):
    return quad_vec(integrand, 0, np.inf, args=(sy, s, M, Z))[0]

# double integral part 2
# quad_vec is used instead of something like dblquad so that 2d arrays of s and M may be calculated efficiently
def integral2(s, M, Z):
    return preFactor*4*quad_vec(integral1, 0, np.inf, args=(s, M, Z))[0]


# basic Gaussian for parameterised fit of f'
def gauss(x, a, b):
    return a*np.exp(-abs(b)*x**2)

# sum of 4 gaussians + constant gives the parameterised fit for f'
def curve(x, *args):
    f = gauss(x, args[0], args[1]) + \
        gauss(x, args[2], args[3]) + \
        gauss(x, args[4], args[5]) + \
        gauss(x, args[6], args[7]) + args[8]
    return f

# linear interpolation between parameterised f' curves, accepts arrays of s
def interpolator(s, B, parameters):
    if np.any(B == Bvalues):
        requiredIndex = np.where(Bvalues == B)[0][0]
        line = curve(s, *parameters[requiredIndex])
        return np.where(line>0, line, 0)
        # return line
    else:    
        requiredIndex = np.where(Bvalues >= B)[0][0]
        requiredB = Bvalues[requiredIndex - 1:requiredIndex + 1]
        line1 = curve(s, *parameters[requiredIndex - 1])
        line2 = curve(s, *parameters[requiredIndex])
        # linear interpolation
        interpolatedLine = line1 + (B - requiredB[0])*(line2 - line1)/(requiredB[1] - requiredB[0])
        # remove points below zero
        # interpolatedLine = np.where(line1*line2 > 0, interpolatedLine, 0)
        interpolatedLine = np.where(interpolatedLine > 0, interpolatedLine, 0)
        return interpolatedLine

# interpolated f' with error checking, accepts arrays of s
def fprime(s, B, Z):
    if isinstance(s, np.ndarray) and np.any(s<0):
        raise Exception("invalid values of s")
    elif isinstance(s, (int, float)) and s<0:
        raise Exception("invalid value of s")
    if B<0:
        raise Exception("invalid value of B")
    if Z<1 or Z>118:
        raise Exception("invalid value of Z")
    if B>4 or 0<B<0.1:
        raise Exception("B outside range of parameterisation")
    if Z>103:
        raise Exception("elements heavier than Z=103 have not been parameterised")
    if isinstance(s, np.ndarray) and B==0:
        return np.zeros(np.shape(s))
    elif isinstance(s, (int, float)) and B==0:
        return 0
    result = interpolator(s, B, parameterArray[Z-1])
    return result

# constants to apply for a given accelerating voltage
V = 200000.0
# lorentz factor
gamma = 1 + (e*V)/(m_e*c**2)
# electron velocity
v = c*np.sqrt(1 - 1/gamma**2)
# prefactor
preFactor = 1e10*2*h/(m_e*c)  # Angstroms


# %% plot of f'*exp(-Bs^2)
Z = 82
# NB Bvalues = np.array([0.1, 0.15, 0.2, 0.25, 0.3, 0.4, 0.5, 0.7, 1, 1.5, 2, 2.75, 4])
B = 2.75
svals = np.linspace(0, 4, 60)

#B&K integral
fprime_i = integral2(svals, B, Z)
fpi_DWF = fprime_i*np.exp(-B*svals*svals)

#parameterised version
requiredIndex = np.where(Bvalues == B)[0][0]
fprime_p = curve(svals, *parameterArray[Z-1][requiredIndex])
fpp_DWF = fprime_p*np.exp(-B*svals*svals)

# set the bottom limit for the plot
mm = -0.
fpi_DWF = np.where(fpi_DWF>mm,fpi_DWF, np.nan)
fpp_DWF = np.where(fpp_DWF>mm,fpp_DWF, np.nan)
fprime_p = np.where(fprime_p>mm,fprime_p, np.nan)
fprime_i = np.where(fprime_i>mm,fprime_i, np.nan)

# plot
plt.rc('font', size=10)
#plt.scatter(svals,fprime_p)
#plt.scatter(svals,fprime_i)
plt.scatter(svals,fpp_DWF)
plt.scatter(svals,fpi_DWF)
plt.grid(color='grey', linestyle='-', linewidth=1)

#%% Max error vs Z plot
maxerror = ([])
z_list = ([])
for Z in range(1,103):
    print(Z)
    svals = np.linspace(0, 3, 60)
    Bvals = np.linspace(0.1, 4, 80)
    X, Y = np.meshgrid(svals, Bvals)
    
    actualZ = integral2(X, Y, Z)
    actualZpos = np.where(actualZ > 0, actualZ, 0)
    interp = []
    for B in Bvals:
        interp.append(fprime(svals, B, Z))
    interZ = np.array(interp)
    # relerror = abs(actualZpos - interZ)/actualZpos
    abserror = abs(actualZpos - interZ)
    fzeros = np.where(actualZ <= 0, 0, np.nan)
    z_list.append(Z)
    maxerror.append(np.max(abserror))

plt.rc('font', size=40)
fig, ax = plt.subplots(figsize=(20,15))
ax.scatter(z_list, maxerror, s=100, c='black', marker='o')
ax.set_xlabel("Atomic number Z")
ax.set_ylabel("Maximum error in $f'$ (Å)")#, fontsize=25)
ax.set_xlim([0, 102])
ax.set_ylim([0, 0.035])
plt.grid(color='grey', linestyle='-', linewidth=2)


# %% calculate array of f', both interpolated and from integral
Z=78
svals = np.linspace(0, 3, 60)
Bvals = np.linspace(0.1, 4, 80)
X, Y = np.meshgrid(svals, Bvals)

# interpolated from parameterised curves
interp = []
for B in Bvals:
    interp.append(fprime(svals, B, Z))
interZ = np.array(interp)
#convert from f' to  f'*exp(-Bs^2)
interZcor = interZ*np.exp(-B*svals*svals)
# from B&K integral
actualZ = integral2(X, Y, Z)
#convert from f' to  f'*exp(-Bs^2)
actualZcor = actualZ*np.exp(-B*svals*svals)
# actualZpos = np.where(actualZ > 0, actualZ, 0)  # only keep positive values

# minfp = 0.01*np.max(actualZ)
# # map of error, only when actualZ positive
# relerror = np.where(actualZ > minfp, abs(actualZ - interZ)/actualZ, 0)
# abserror = np.where(actualZ > minfp, abs(actualZ - interZ), 0)
# # abserror = np.where(actualZ*interZ>0, abserror, 0)
# fzeros = np.where(actualZ <= 0, 0, np.nan)
min = 0
#minfp = 0.01*np.max(actualZcor)
# map of error, only when actualZ positive
relerror = np.where(actualZcor > min, abs(actualZcor - interZcor)/actualZcor, 0)
abserror = np.where(actualZcor > min, abs(actualZcor - interZcor), 0)
# abserror = np.where(actualZ*interZ>0, abserror, 0)
fzeros = np.where(actualZ <= 0, 0, np.nan)

# % % plot f' error map

plt.rc('font', size=35)
c_map = 'CMRmap'

# fig, ax = plt.subplots(figsize=(20,15))
# # ax.set_title("Relative error of parameterisation for gold")#, fontsize=15)
# ax.set_xlabel("s (Å⁻¹)")#, fontsize=25)
# ax.set_ylabel("B (Ų)")#, fontsize=25)
# ax.set_xlim([0, 3])
# ax.set_ylim([0.1, 4])
# plot = ax.pcolormesh(X, Y, relerror, cmap=c_map, vmin=0.0, vmax=0.1)
# ax.pcolormesh(X, Y, fzeros, cmap=c_map)
# fig.colorbar(plot, label = "relative error")


fig, ax = plt.subplots(figsize=(20,15))
# ax.set_title("Absolute error of parameterisation for gold")#, fontsize=15)
ax.set_xlabel("s (Å⁻¹)")#, fontsize=25)
ax.set_ylabel("B (Ų)")#, fontsize=25)
ax.set_xlim([0, 3])
ax.set_ylim([0.1, 4])
plot = ax.pcolormesh(X, Y, abserror, cmap=c_map)#, vmin=0.01, vmax=0.05)
ax.pcolormesh(X, Y, fzeros, cmap=c_map)
fig.colorbar(plot, label = "absolute error")

#%%  write a set of parameters for a latex document
Z=6
Mvals = [0.1, 0.15, 0.2, 0.25, 0.3, 0.4, 0.5, 0.7, 1, 1.5, 2, 2.75, 4]
fel = open('table.txt', 'w')
for j in range(13):
    fel.write(' & ' + str(Mvals[j]))
    for i in range(9):
        fel.write(' & ' + '%.4E' % parameterArray[Z-1, j, i]) 
    fel.write(' \\\\ \n')
fel.write(' & & & & & & & &  \\\\ \n')
fel.write(' \hline')
# fel.write( np.array2string(parameterArray[0, 12,:], precision=4, separator=' &'))
fel.close()