Notebook of Examples for PhotonAttenuation2 Package
This file contains tests, examples and plots related to different functions in PhotonAttenuation2 Package
Contents
- Plot Photon Mass Attenuation Coefficients and Compton Edges
- Plot Log of Mass Energy-Absorption Coefficients
- Plot Photon Attenuation Coefficients for Uranium
- Plot Photon Attenuation Coefficients, using different input styles
- Plot mean free path of photons in different media
- Plot Cross sections of elements for different energy photons
- Hardening of Bremsstrahlung Spectum by Steel
- Dual x-ray ability to recognize different materials, using Mono-energetic energies and Flat Spectra.
- Get properties of concrete
- Access Element Properties through PhysProps
- Access List of Compound names recognized by PhysProps, PhotonAttenuation and ParseChemicalFormula functions
- Test ParseChemicalFormula
- Test consistency between stored and calculated Z/A values
Plot Photon Mass Attenuation Coefficients and Compton Edges
Plot as a function of energy and atomic number of elements See http://physics.nist.gov/PhysRefData/XrayMassCoef/chap2.html for details
figure Z = 1:100; % elements with Z in 1-100 range E = logspace(log10(0.001), log10(20), 500); % define energy grid [mac, CEdge] = PhotonAttenuationQ(Z, E); imagesc(log10(mac)); colorbar; title('Log of Photon Mass Attenuation Coefficients (in cm^2/g) and Compton Edges'); xlabel('Atomic Number of Elements'); ylabel('Energy in MeV'); zlabel('Attenuation in cm^2/g'); set(gca,'YTick',linspace(1, length(E), 10)); set(gca,'YTickLabel',1e-3*round(1e3*logspace(log10(0.001), log10(20), 10))) hold on ed = accumarray([CEdge(:,1),CEdge(:,2)],CEdge(:,3)); % get per element energies of 14 compton edges ed = 500*(log(ed')-log(0.001))/(log(20)-log(0.001)); % convert energy to row numbers of the image plot(ed ,'LineWidth',4); % plot compton edges L = {'K','L1','L2','L3','M1','M2','M3','M4','M5','N1','N2','N3','N4','N5'}; legend(L, 'Location', 'southwest'); % add legend hold off;
Warning: Log of zero.
Plot Log of Mass Energy-Absorption Coefficients
Plot as a function of energy and atomic number of elements See http://physics.nist.gov/PhysRefData/XrayMassCoef/chap3.html for details
figure Z = 1:92; % elements with Z in 1-92 range (Elements higher than 92 are not defined) E = logspace(log10(0.001), log10(20), 500); % define energy grid meac = PhotonAttenuationQ(Z, E, 'meac'); imagesc(log10(meac)); colorbar; title('Log of Photon Mass Energy-Absorption Coefficients in cm^2/g'); xlabel('Atomic Number of Elements'); ylabel('Energy in MeV'); set(gca,'YTick',linspace(1, length(E), 10)); set(gca,'YTickLabel',1e-3*round(1e3*logspace(log10(0.001), log10(20), 10)))
Plot Photon Attenuation Coefficients for Uranium
Compare with http://physics.nist.gov/PhysRefData/XrayMassCoef/ElemTab/z92.html
figure E = logspace(log10(0.001), log10(20), 500); % define energy grid mac = PhotonAttenuationQ(92, E, 'mac'); meac = PhotonAttenuationQ(92, E, 'meac'); loglog(E, mac); hold on; loglog(E, meac, 'b-.'); legend({'mac', 'meac'}); ylabel('Attenuation in cm^2/g'); xlabel('Photon Energy in MeV'); title({'Photon Attenuation Coefficients for Uranium',... 'see http://physics.nist.gov/PhysRefData/XrayMassCoef/ElemTab/z92.html'});
Plot Photon Attenuation Coefficients, using different input styles
figure E = logspace(log10(0.001), log10(20), 500); % define energy grid Z = {'Concrete', 'Air', 'B(10)H(11)C(58)O(21)', 100, 'Ag'}; mac = PhotonAttenuation(Z, E, 'mac'); loglog(E, mac); legend({'Concrete', 'Air', 'BPE-10', 'Fermium', 'Silver'}); ylabel('Attenuation in cm^2/g'); xlabel('Photon Energy in MeV'); title('Photon Attenuation Coefficients for different materials');
Plot mean free path of photons in different media
Plot as function of energy and atomic number of media/element
figure Z = 1:99; % elements with Z in 1-99 range (Fermium Z=100 was dropped since its density is not known) E = -logspace(log10(6.198e-5), log10(1.2395), 500); % wavelength instead of energy X = PhotonAttenuation(Z, E, 'mean free path'); imagesc(log10(X)); colorbar; title({'Log of Mean-free-path of photons in different media (in cm)', ... '(Vertical bars are due to low density of gases)'}); xlabel('Atomic Number of Elements'); ylabel('Photon wavelength in nm'); set(gca,'YTick',linspace(1, length(E), 10)); set(gca,'YTickLabel',1e-4*round(1e4*logspace(log10(6.198e-5), log10(1.2395), 10)))
Plot Cross sections of elements for different energy photons
figure Z = 1:100; % elements with Z in 1-100 range E = logspace(log10(0.001), log10(20), 500); % define energy grid X = PhotonAttenuation(Z, E, 'cross section'); imagesc(log10(X)); colorbar; title({'Log of Cross sections of elements for different energy photons',... ' In barns/atom or in 10E-24 cm^2'}); xlabel('Atomic Number of Elements'); ylabel('Photon Energy in MeV'); set(gca,'YTick',linspace(1, length(E), 10)); set(gca,'YTickLabel',1e-3*round(1e3*logspace(log10(0.001), log10(20), 10)))
Hardening of Bremsstrahlung Spectum by Steel
Spectra change differently as they pass through different materials
figure %Define 9 MeV Bremsstrahlung Spectrum E=[0.05,0.0506,0.0664,0.0873,0.1147,0.1506,0.1979,0.2599,0.3414,0.4485,... 0.5891,0.7738,1.0165,1.3352,1.7539,2.3038,3.0262,3.9751,5.2215,6.8587,8,9]'; S=[1,3,6,23,90,175,280,365,400,415,392,374,358,326,281,223,175,119,81,44,20,1]'; T_Al = PhotonAttenuation('Aluminum', E, 'transmission', 5); T_Pb = PhotonAttenuation('Lead' , E, 'transmission', 1); S_Al = repmat(S,1,25); S_Pb = repmat(S,1,25); for i=2:10 S_Al(:,i) = S_Al(:,i-1).*T_Al; S_Pb(:,i) = S_Pb(:,i-1).*T_Pb; end subplot(2,1,1) loglog(E,S_Al); xlim([0.05, 9]); ylim([1, 500]); ylabel('Number of photons'); title({'Hardening of Bremsstrahlung Spectum by Aluminum.' 'Each line coresponds to 5 cm of Al'}) subplot(2,1,2) loglog(E,S_Pb); xlim([0.05, 9]); ylim([1, 500]); xlabel('Photon Energy in MeV'); ylabel('Number of photons'); title({'Hardening of Bremsstrahlung Spectum by Lead.' 'Each line coresponds to 1 cm of lead'})
Dual x-ray ability to recognize different materials, using Mono-energetic energies and Flat Spectra.
Notice that ratio of logs of transmitions for 10 MeV and 5 MeV photons is depent on thickness of the material in case of flat spectras, but is independent of thickness in case of mono-energetic energies
E0 = 0.5; EH = exp(log(E0):0.005:log(10)); % define spectrum range EL = exp(log(E0):0.005:log(5)); EH = [EH; ones(1,length(EH))]; % define flat spectrum EL = [EL; ones(1,length(EL))]; T = logspace(0,log10(400),50); % mass thickness in g/cm^2 Z = 1:100; % elements with Z in 1-100 range TL1 = zeros(length(T),length(Z)); TH1 = TL1; TL2 = TL1; TH2 = TL1; for i = 1:length(T) TL1(i,:) = -log(PhotonAttenuation(Z, 5, 'Transmission', -T(i))); TH1(i,:) = -log(PhotonAttenuation(Z, 10, 'Transmission', -T(i))); TL2(i,:) = -log(PhotonAttenuation(Z, EL, 'Transmission', -T(i))); TH2(i,:) = -log(PhotonAttenuation(Z, EH, 'Transmission', -T(i))); end ratio1 = TH1./TL1; ratio2 = TH2./TL2; [tmp i]=min(ratio1(:)); ratio2( 1 )=ratio1(i); % make sure ranges ... [tmp i]=max(ratio1(:)); ratio2(end)=ratio1(i); % .. are the same figure('Position',[1 1 800 600]) subplot(1,2,2); imagesc(ratio1'); colorbar; title('Mono-energetic 5 & 10 MeV sources.'); ylabel('Atomic Number Z'); xlabel('Mass Thickness in g/cm^2'); t = 1:8:50; set(gca,'XTick',t); lab = cell (1,length(t)); for i = 1:length(t), lab{i} = num2str(round(T(t(i)))); end set(gca,'XTickLabel',lab); subplot(1,2,1); imagesc(ratio2'); %colorbar; title('Flat spectra Min=0.5, Max=5 & 10 MeV.'); xlabel('Mass Thickness in g/cm^2'); set(gca,'XTick',t); set(gca,'XTickLabel',lab);
Get properties of concrete
Show how to use PhysProps, ParseChemicalFormula & PhotonAttenuation functions for accesing various physical properties of materials
X = PhysProps('Concrete'); Concrete.Density = X{2}; % material density Concrete.Composition = X{3}; % element composition Concrete.Z_A = X{1}; % mean atomic number to atomic mass ratio [Z R] = ParseChemicalFormula(X{3}); MFP = PhotonAttenuation(X{3}, 0.662, 'mean free path'); Concrete.ElementZ = Z'; % atomic numbers of elements Concrete.ElementRatio = R'; % weight ratio of elements Concrete.MeanFreePath = MFP; % Mean Free Path of gammas from Cs-137 source disp(Concrete) % display the data
Density: 2.3000
Composition: [1x113 char]
Z_A: 0.5093
ElementZ: [1 6 8 11 12 13 14 19 20 26]
ElementRatio: [0.0221 0.0025 0.5749 0.0152 0.0013 0.0200 0.3046 0.0100 0.0430 0.0064]
MeanFreePath: 5.5160
Access Element Properties through PhysProps
P = PhysProps('Element Data'); [{'Atomic Number', 'Element Symbol', 'Z/A'} ; num2cell((1:100)'), P]
ans =
'Atomic Number' 'Element Symbol' 'Z/A'
[ 1] 'H' [0.9921]
[ 2] 'He' [0.4997]
[ 3] 'Li' [0.4322]
[ 4] 'Be' [0.4438]
[ 5] 'B' [0.4625]
[ 6] 'C' [0.4995]
[ 7] 'N' [0.4998]
[ 8] 'O' [0.5000]
[ 9] 'F' [0.4737]
[ 10] 'Ne' [0.4955]
[ 11] 'Na' [0.4785]
[ 12] 'Mg' [0.4937]
[ 13] 'Al' [0.4818]
[ 14] 'Si' [0.4985]
[ 15] 'P' [0.4843]
[ 16] 'S' [0.4990]
[ 17] 'Cl' [0.4795]
[ 18] 'Ar' [0.4506]
[ 19] 'K' [0.4859]
[ 20] 'Ca' [0.4990]
[ 21] 'Sc' [0.4671]
[ 22] 'Ti' [0.4595]
[ 23] 'V' [0.4515]
[ 24] 'Cr' [0.4616]
[ 25] 'Mn' [0.4551]
[ 26] 'Fe' [0.4656]
[ 27] 'Co' [0.4582]
[ 28] 'Ni' [0.4771]
[ 29] 'Cu' [0.4564]
[ 30] 'Zn' [0.4588]
[ 31] 'Ga' [0.4446]
[ 32] 'Ge' [0.4407]
[ 33] 'As' [0.4405]
[ 34] 'Se' [0.4306]
[ 35] 'Br' [0.4380]
[ 36] 'Kr' [0.4296]
[ 37] 'Rb' [0.4329]
[ 38] 'Sr' [0.4337]
[ 39] 'Y' [0.4387]
[ 40] 'Zr' [0.4385]
[ 41] 'Nb' [0.4413]
[ 42] 'Mo' [0.4378]
[ 43] 'Tc' [0.4392]
[ 44] 'Ru' [0.4353]
[ 45] 'Rh' [0.4373]
[ 46] 'Pd' [0.4323]
[ 47] 'Ag' [0.4357]
[ 48] 'Cd' [0.4270]
[ 49] 'In' [0.4268]
[ 50] 'Sn' [0.4212]
[ 51] 'Sb' [0.4189]
[ 52] 'Te' [0.4075]
[ 53] 'I' [0.4176]
[ 54] 'Xe' [0.4113]
[ 55] 'Cs' [0.4138]
[ 56] 'Ba' [0.4078]
[ 57] 'La' [0.4103]
[ 58] 'Ce' [0.4139]
[ 59] 'Pr' [0.4187]
[ 60] 'Nd' [0.4160]
[ 61] 'Pm' [0.4209]
[ 62] 'Sm' [0.4123]
[ 63] 'Eu' [0.4146]
[ 64] 'Gd' [0.4070]
[ 65] 'Tb' [0.4090]
[ 66] 'Dy' [0.4062]
[ 67] 'Ho' [0.4062]
[ 68] 'Er' [0.4066]
[ 69] 'Tm' [0.4084]
[ 70] 'Yb' [0.4045]
[ 71] 'Lu' [0.4058]
[ 72] 'Hf' [0.4034]
[ 73] 'Ta' [0.4034]
[ 74] 'W' [0.4025]
[ 75] 'Re' [0.4028]
[ 76] 'Os' [0.3996]
[ 77] 'Ir' [0.4006]
[ 78] 'Pt' [0.3998]
[ 79] 'Au' [0.4011]
[ 80] 'Hg' [0.3988]
[ 81] 'Tl' [0.3963]
[ 82] 'Pb' [0.3957]
[ 83] 'Bi' [0.3972]
[ 84] 'Po' [0.4019]
[ 85] 'At' [0.4048]
[ 86] 'Rn' [0.3874]
[ 87] 'Fr' [0.3901]
[ 88] 'Ra' [0.3893]
[ 89] 'Ac' [0.3920]
[ 90] 'Th' [0.3879]
[ 91] 'Pa' [0.3939]
[ 92] 'U' [0.3865]
[ 93] 'Np' [0.3923]
[ 94] 'Pu' [0.3851]
[ 95] 'Am' [0.3908]
[ 96] 'Cm' [0.3886]
[ 97] 'Bk' [0.3926]
[ 98] 'Cf' [0.3903]
[ 99] 'Es' [0.3927]
[ 100] 'Fm' [0.3890]
Access List of Compound names recognized by PhysProps, PhotonAttenuation and ParseChemicalFormula functions
P = PhysProps('Compound Names'); P = [{'Name', 'Alternative Name', 'Molecular Formula'} ; P(:,[2 1 3])]; idx = find( cellfun(@length, P(:,1)) < 20 ); display(P(idx,:)) % display only names shorter than 20 characters
'Name' 'Alternative Name' 'Molecular Formula'
'Aluminium' 'Al' 'Al'
'Sulphur' 'S' 'S'
'Caesium' 'Cs' 'Cs'
'Graphite' 'C' 'C'
'Diamond' 'C' 'C'
'Carbon, Amorphous' 'C' 'C'
'ACETONE' '' 'C3H6O'
'ACETYLENE' '' 'C2H2'
'ADENINE' '' 'C5H5N5'
'ALANINE' '' 'C3H7NO2'
'ALUMINUM OXIDE' '' 'Al2O3'
'AMBER' '' 'C10H16O'
'AMMONIA' '' 'NH3'
'ANILINE' '' 'C6H7N'
'ANTHRACENE' '' 'C14H10'
'BAKELITE' '' [1x33 char]
'BARIUM FLUORIDE' '' 'BaF2'
'BARIUM SULFATE' '' 'BaSO4'
'BENZENE' '' 'C6H6'
'BERYLLIUM OXIDE' '' 'BeO'
'BLOOD (ICRP)' 'Blood' [1x161 char]
'BORON CARBIDE' '' 'B4C'
'BORON OXIDE' '' 'B2O3'
'BRAIN (ICRP)' 'Brain' [1x149 char]
'BUTANE' '' 'C4H10'
'N-BUTYL ALCOHOL' 'n-BUTANOL' 'C4H10O'
'CADMIUM TELLURIDE' '' 'CdTe'
'CADMIUM TUNGSTATE' 'CWO' 'CdWO4'
'CALCIUM CARBONATE' '' 'CaCO3'
'CALCIUM FLUORIDE' '' 'CaF2'
'CALCIUM OXIDE' '' [1x23 char]
'CALCIUM SULFATE' '' 'CaSO4'
'CALCIUM TUNGSTATE' '' 'CaWO4'
'CARBON DIOXIDE' '' 'CO2'
'CELLULOSE NITRATE' 'Nitrocellulose' 'C6H8N2O9'
'CESIUM FLUORIDE' '' 'CsF'
'CESIUM IODIDE' '' 'CsI'
'CHLOROBENZENE' '' 'C6H5Cl'
'CHLOROFORM' '' 'CHCl3'
'CONCRETE, PORTLAND' 'Portland Concrete' [1x116 char]
'Concrete, Ordinary' 'Concrete' [1x113 char]
'Concrete, Barite' 'Barite Concrete' [1x103 char]
'CYCLOHEXANE' '' 'C6H12'
'1, 2-DICHLOROETHANE' 'EDC' 'C2H4Cl2'
'DIETHYL ETHER' '' 'C4H10O'
'DIMETHYL SULFOXIDE' 'DMSO' 'C2H6OS'
'ETHANE' '' 'C2H6'
'ETHYL ALCOHOL' 'Ethanol' 'C2H6O'
'ETHYL CELLULOSE' '' 'C12H22O5'
'ETHYLENE' '' 'C2H4'
'EYE LENS (ICRP)' 'EYE LENS' [1x44 char]
'FERRIC OXIDE' 'Iron(III) oxide' 'Fe2O3'
'FERROBORIDE' '' 'FeB'
'FERROUS OXIDE' 'Iron(II) oxide' 'FeO'
'FREON-12' [1x23 char] 'CCl2F2'
'FREON-12B2' [1x22 char] 'CBr2F2'
'FREON-13' '' 'CClF3'
'FREON-13B1' '' 'CBrF3'
'FREON-13I1' '' 'CIF3'
'GALLIUM ARSENIDE' '' 'GaAs'
'Glass, Borosilicate' 'Pyrex Glass' [1x69 char]
'GLASS, LEAD' 'Lead Glass' [1x59 char]
'GLASS, PLATE' 'Plate Glass' [1x47 char]
'GLUCOSE' 'GLc' 'C6H14O7'
'GLUTAMINE' '' 'C5H10N2O3'
'GLYCEROL' 'Glycerin' 'C3H8O3'
'GUANINE' '' 'C5H5N5O'
'N-HEPTANE' 'HEPTANE' 'C7H16'
'N-HEXANE' 'HEXANE' 'C6H14'
'LEAD OXIDE' '' 'PbO'
'LITHIUM AMIDE' '' 'LiNH2'
'LITHIUM CARBONATE' '' 'Li2CO3'
'LITHIUM FLUORIDE' '' 'LiF'
'LITHIUM HYDRIDE' '' 'LiH'
'LITHIUM IODIDE' '' 'LiI'
'LITHIUM OXIDE' '' 'Li2O'
'LITHIUM TETRABORATE' '' 'Li2B4O7'
'LUNG (ICRP)' 'Lung' [1x149 char]
'M3 WAX' '' [1x57 char]
'MAGNESIUM CARBONATE' '' 'MgCO3'
'MAGNESIUM FLUORIDE' '' 'MgF2'
'MAGNESIUM OXIDE' '' 'MgO'
'MERCURIC IODIDE' '' 'HgI2'
'METHANE' '' 'CH4'
'METHANOL' '' 'CH4O'
'MIX D WAX' '' [1x57 char]
'NAPHTHALENE' '' 'C10H8'
'NITROBENZENE' '' 'C6H5NO2'
'NITROUS OXIDE' '' 'N2O'
'NYLON, TYPE 6/10' '' 'C8H15ON'
'OCTANE, LIQUID' 'Liquid Octane' 'C8H18'
'Ovary' '' [1x74 char]
'PARAFFIN WAX' 'PARAFFIN' 'C25H52'
'N-PENTANE' 'PENTANE' 'C5H12'
'PLUTONIUM DIOXIDE' '' 'PuO2'
'POLYACRYLONITRILE' 'PAN' 'C3H3N'
'POLYCHLOROSTYRENE' '' 'C17H18Cl2'
'POLYETHYLENE' '' 'C2H4'
'POLYOXYMETHYLENE' '' 'H2CO'
'POLYPROPYLENE' '' 'C3H6'
'POLYSTYRENE' '' 'CH'
'POLYVINYL ACETATE' 'PVA' 'C4H6O2'
'POLYVINYL ALCOHOL' 'PVOH' 'C2H4O'
'POLYVINYL BUTYRAL' 'PVB' 'C8H13O2'
'POLYVINYL CHLORIDE' 'PVC' 'C2H3Cl'
'POTASSIUM IODIDE' '' 'KI'
'POTASSIUM OXIDE' '' 'K2O'
'PROPANE' '' 'C3H8'
'PROPANE, LIQUID' 'LIQUID PROPANE' 'C3H8'
'N-PROPYL ALCOHOL' '1-Propanol' 'C3H8O'
'PYRIDINE' 'Azine' 'C5H5N'
'RUBBER, BUTYL' 'BUTYL' 'C4H8'
'RUBBER, NATURAL' 'Latex' 'C5H8'
'RUBBER, NEOPRENE' 'NEOPRENE' 'C4H5Cl'
'SILICON DIOXIDE' '' 'SiO2'
'SILVER BROMIDE' '' 'AgBr'
'SILVER CHLORIDE' '' 'AgCl'
'SILVER IODIDE' '' 'AgI'
'SKIN (ICRP)' 'Skin' [1x149 char]
'SODIUM CARBONATE' 'Washing soda' 'Na2CO3'
'SODIUM IODIDE' '' 'NaI'
'SODIUM MONOXIDE' '' 'Na2O'
'SODIUM NITRATE' '' 'NaNO3'
'STILBENE' [1x27 char] 'C14H12'
'SUCROSE' 'Sugar' 'C12H22O11'
'TERPHENYL' 'p-Terphenyl' 'C18H10'
'TESTES (ICRP)' 'TESTES' [1x149 char]
'TETRACHLOROETHYLENE' 'PCE' 'C2Cl4'
'THALLIUM CHLORIDE' '' 'TlCl'
'TISSUE, SOFT (ICRP)' 'Soft Tissue' [1x149 char]
'TITANIUM DIOXIDE' '' 'TiO2'
'TOLUENE' 'methylbenzene' 'C7H8'
'TRICHLOROETHYLENE' 'TCE' 'Cl3C2H'
'TRIETHYL PHOSPHATE' 'phosphoric acid' 'C6H15PO4'
'URANIUM DICARBIDE' '' 'UC2'
'URANIUM MONOCARBIDE' 'Uranium carbide' 'UC'
'URANIUM OXIDE' 'Uranium dioxide' 'UO2'
'UREA' 'Diaminomethanal' 'N2H4CO'
'VALINE' 'Val' 'C5H11NO2'
'WATER, LIQUID' 'Water' 'H2O'
'WATER VAPOR' 'Steam' 'H2O'
'WATER, ICE' 'Ice' 'H2O'
'XYLENE' '' 'C8H10'
'Yellowcake' '' 'U3O8'
'Steel, ASTM A 366' 'Steel' [1x46 char]
'' '' ''
To See the full list see CompoundNames.html file
Test ParseChemicalFormula
Run and make sure nothing crashes
ParseChemicalFormula('Pb'); % Element Symbol ParseChemicalFormula('Lead'); % Element Name ParseChemicalFormula('Water'); % Common name ParseChemicalFormula('H2SO4'); % Molecular formula ParseChemicalFormula('CO2(3)CO(5)'); % Mix of Molecular formulas Bakelite = 'H(0.057444) C(0.774589) O(0.167968)'; ParseChemicalFormula(Bakelite); % Mix of Elements using fractions GafChromic = 'H(0.0897) C(0.6058) N2O3(0.3045)'; ParseChemicalFormula(GafChromic); % Mix of Molecular formulas BPE10 = 'B(10)H(11)C(58)O(21)'; ParseChemicalFormula(BPE10); % Mix of Elements using ratios
Test consistency between stored and calculated Z/A values
P = PhysProps('All Data'); ZA = [P{:,1}]'; % extract ZA of all materials za = zeros(size(ZA)); for i=1:(size(P,1)-1) [Z R] = ParseChemicalFormula(P{i,3}); za(i) = dot(R,ZA(Z)); end fprintf('Maximum discrepancy between calculated and stored ZA = %f\n', max(abs(za-ZA)));
Maximum discrepancy between calculated and stored ZA = 0.075890