easyspin

Download and requirementsEasySpin comes in a single zip file, containing all toolbox functions and the full documentation.Visit the EasySpin website for downloading the latest version.EasySpin requires MATLAB 7.0 (R14) or later on either Windows, Linux or Mac. Beyond a basic MATLAB installation, no additional MATLAB toolboxes are needed to run EasySpin.Installation1.Download and install support softwareA small but critical part of EasySpin is not written in MATLAB, but in C. In order tocompile and run this part, some support software has to be installed, depending on theplatform.o Mac OS X: Download and install Xcode 3 (not Xcode 4).o64-bit Windows: Download and install Microsoft Visual C++ 2010Redistributable Package (x64).o32-bit Windows or Linux: Skip this step.2.Download and unpack EasySpinDownload the EasySpin zip file and unpack it to a folder, e.g., C:\ or/var/myfiles/. Once unpacked, EasySpin is contained in a subfolder of C:\ (or/var/myfiles/ or whatever directory you chose) which in turn contains varioussubfolders:o easyspin - all the toolbox functions.o documentation - documentation, entry point is index.html.o examples - all examples, grouped into subdirectories.3.Tell MATLAB about EasySpinLaunch MATLAB and go to menu File → Set Path... Remove any folders containing olderEasySpin installations from the MATLAB search path. Add the EasySpin subfoldereasyspin to the MATLAB search path by clicking on "Add Folder...", selecting theeasyspin subfolder in your EasySpin directory, and clicking on "Save".pile EasySpinIn MATLAB, type easyspin at the command prompt. Possibly you are asked to selecta C compiler. In that case, choose Lcc (Windows 32bit) or gcc (Linux, Mac). The finaloutput in the command window should look like5.==================================================================6.7. EasySpin, a MATLAB toolbox for ElectronParamagnetic Resonance8.9. Release: 4.0.0 (2011-10-16)10.11.==================================================================12.13. Expiry date: 30-Jun-201314. Directory:D:\easyspin-head\easyspin15. MATLAB version: 7.13.0.564 (R2011b)16. Platform: Microsoft Windows [Version6.1.7601]17. System date: 16-Oct-2011 16:12:5718. Temp dir:C:\Users\abc\AppData\Local\Temp\19. mex-files: mexw64, ok20.21.==================================================================22.DocumentationTo view the documentation, point your web browser todocumentation/index.html in your EasySpin installation directory andbookmark that page. In addition, EasySpin documentation is also accessible via theMATLAB help browser. It is listed together with other installed toolboxes. Type doc inthe MATLAB command window to access the MATLAB help system.License1.You may download and use EasySpin free of charge and without any restrictions.2.You may copy and distribute verbatim copies of EasySpin.3.You may not rent or sell any part of EasySpin.4.You may not use or modify EasySpin or a part of it for other software which is not freelyavailable at no cost.5.You may not reverse engineer, decompile or disassemble EasySpin.6.EasySpin comes without warranty of any kind.7.If you use results obtained with the help of EasySpin in any scientific publication, cite theappropriate publication.About EasySpinEasySpin is a MATLAB toolbox for simulating and fitting Electron Paramagnetic Resonance (EPR) spectra. It supplements the numerical and visualisation power of MATLAB with the best computational methods devisedby EPR spectroscopists. EasySpin runs on Windows, Linux and Mac, and is available free of charge.Current version: 4.0.0 (16 Oct 2011) [download]FeaturesEasySpin can simulate a wide range of EPR spectra. For all simulations, interactive least-squares fitting using hybrid methods based on Simplex,Levenberg-Marquardt and genetic algorithms are possible.Solid-state cw EPR spectra∙powders and crystals (with space groups) ∙arbitrary number of electron and nuclear spins ∙all interactions, including high-order operators and nuclear quadrupole ∙matrix diagonalization and perturbation methods ∙broadening models: g, A and D strain, unresolved hyperfine splittings ∙non-equilibrium populations ∙ perpendicular and parallel detection modePulse EPR∙two- and three-pulse ESEEM, HYSCORE ∙user-defined sequences ∙arbitrary number of nuclear spins ∙high-spin systems ∙nuclear quadrupole interaction included ∙ built-in processingSlow-motion cw EPR spectra∙one unpaired electron, several nuclei ∙axial and rhombic rotational diffusion tensor ∙arbitrary tensor orientations ∙orientational potential ∙ single-orientation and MOMD modelsSolid-state ENDOR spectra∙powders and crystals (with space groups)∙arbitrary number of nuclei, includes nuclearquadrupole∙hyperfine enhancement∙built-in orientation selection∙matrix diagonalization and perturbation methodsIsotropic cw EPR spectra∙one unpaired electron, arbitrary number of nuclei∙resonance fields are exact (no perturbation formulae)∙automatic determination of magnetic field rangeFast-motion cw EPR spectra∙one unpaired electron, arbitrary number of nuclei∙user provides rotational correlation time and tensoranisotropies, line widths are automatically computed∙includes effects from nuclear quadrupole interaction EasySpin also provides a variety of other tools:∙Basic spin physics: spin operators, Stevens operators, spin Hamiltonians with arbitrary number of electron and nuclear spins, angular momentum.∙Utilities: line shapes, Euler angle utilities, orientational grids, data import from all common EPR file formats (BES3T, ESP, Varian), nuclear isotope database,field/frequency/g value conversion∙Data analysis: RC filtering, field-modulation, cross-term averaged FFT, moving average, many apodization windows, baseline correction∙Resonances: ENDOR and cw EPR resonance line computations, energy level diagrams ∙Time-domain: support for propagators and density matrix evolutions.EasySpin referenceBasics:∙Defining a spin system∙Line broadeningsSpectral simulations and least-squares fitting:∙garlic: cw EPR (isotropic and fast motion) user guide∙chili: cw EPR (slow motion) user guide∙pepper: cw EPR (solid state) user guide∙salt: ENDOR (solid state) user guide∙saffron: pulse EPR/ENDOR (solid state)∙esfit: least-squares fitting user guideThere is also a collection of examples.List of all EasySpin functions: alphabetically, by category.New features and changes, from release to releaseHow to install EasySpin∙EPR toolsSpin operators and matricesSpin HamiltonianHigh-order spin operatorsLinewidths in the fast-motion regimeRotations and Euler anglesLine shapesNuclear isotope table∙Defining a spin systemEasySpin uses mainly two data types that are available in MATLAB: arrays and structures. In EasySpin, a spin system is represented by a MATLAB structure (the spin system structure) containing certain fields.The spin system structure contains all information about the spin system and its spin Hamiltonian. Its fields specify spin quantum numbers, interaction parameters, matrices and tensors, relative orientation angles for these matrices and tensors as well as details about broadenings.The spin Hamiltonian is set up in frequency units (MHz, not angular frequencies), so all the energy parameters of the Hamiltonian have to begiven in MHz as well. Hence, for example, the field A represents in factthe diagonal of the hyperfine coupling tensor divided by the Planck constant, A/h, and not of A. Remember that all angles are in radians, not in degrees.Spin system structures are used by many EasySpin functions, among them the simulation functions chili (slow-motion EPR), garlic (isotropic EPR), pepper(solid-state EPR), salt(ENDOR), and saffron(pulse EPR).A few examplesBefore we give a full list of possible fields, here are a couple of examples of spin system definitions. There are two equivalent ways.Sys.S = 1/2;Sys.g = [1.9 2.0 2.1];Sys.lw = 0.7; % mTSys = struct('S',1/2,'g',[1.9 2.0 2.1],'lw',0.7);Nuclei can be added to a spin system either using a set of fields (Nucs, A) or using the function nucspinadd, which is more convenient in many situations.Sys.S = 1/2;Sys.g = [2 2.1 2.2];Sys.Nucs = '63Cu';Sys.A = [50 50 460];Sys = struct('S',1/2,'g',[2 2.1 2.2]);Sys = nucspinadd(Sys,'63Cu',[50 50 460]);A high-spin Mn2+ system with zero-field splitting isSys = struct('S',5/2,'g',2,'Nucs','55Mn');Sys.A = -240;Sys.D = 120;Parameter referenceThe following groups of parameters can be specified in the spin system structure:∙Spins∙g values∙Hyperfine couplings∙Nuclear quadrupole couplings∙Zero-field splittings∙Higher-order operators∙Electron-electron couplings∙BroadeningsAll interaction matrices and tensors can have arbitrary orientations with respect to a molecule-fixed frame of reference (the so-called molecular frame).Below we list and describe all possible spin system structure fields containing spin Hamiltonian parameters.The spinsThe two fields S and Nucs are used to specify the electron and nuclearspins constituting the spin system. Both are optional. If S is not given,S=1/2 is assumed. If Nucs is not specified, Nucs='' is used.S Gives the electron spin quantum number(s). An arbitrary number of electron spins can be specified. Examples:Sys.S = 1/2; % one electron spinwith S=1/2Sys.S = 5/2; % an S=5/2 spinSys.S = [1/2, 1/2]; % two S=1/2 spinsSys.S = [1, 1, 1/2]; % two S=1 spins and oneS=1/2 spinIf S is not given, it is automatically set to 1/2.Nucs A string containing a comma-separated list of nuclear isotopes specifying the nuclear spins in the spin system. An arbitrarynumber of nuclei can be specified.Sys.Nucs = '1H'; % one hydrogenSys.Nucs = '63Cu'; % a 63Cu nucleusSys.Nucs = '59Co,14N,14N'; % a 59Co and two14N nucleiIf there are no nuclear spins in the system, don't specify this field or set it to '' (an empty string). Nuclei can be added and removed one at a time using nucspinadd and nucspinrmv.If not a single isotope, but a natural-abundance mixture of isotopes is needed, just omit the mass number. You can also freely mix single isotopes and natural-abundance mixtures in one spin system.Sys.Nucs = 'Cu'; % natural-abundancemixture of 69% 63Cu and 31% 65CuSys.Nucs = 'Cu,14N'; % same as above,plus a pure 14NSys.Nucs = 'F,C'; % 100% 19F, plus amixture of 99% 12C and 1% 13CEasySpin will automatically simulate the spectra of all possible isotopologues (isotopes combinations) and combine them with the appropriate weights. Hyperfine constants and quadrupole couplings are automatically converted between isotopes. The values given by you in the spin system are taken to apply for the most abundant isotope with an appropriate spin (e.g. 63Cu in the case of Cu, 1H for H, 14N of N, 119Sn for Sn).It is also possible to give custom/enriched isotope mixtures by explicitly giving a list of the mass numbers of all isotopesin parentheses in Nucs. Additionally, the abundances should be given in Abund. In the simplest case of one atom, this would beSys.Nucs = '(12,13)C'; % for a mixture of12C and 13CSys.Abund = [0.3 0.7]; % 30% of 12C, therest 13CSys.Nucs = '(1,2)H'; % for a mixture ofprotons and deuteronsSys.Abund = [0.05 0.95]; % 95% deuteriumIn the case of multiple atoms, Abund should be a cell arraywith a list of abundances for each atom. For exampleSys.Nucs = '63Cu,(32,33)S'; % 63Cu withenriched 33SSys.Abund = {1,[0.1,0.9]]; % 100% 63Cu,10% 32S and 90% 33SCustom and natural abundance mixtures and single isotopes canbe all used at the same time. Any entry for a natural-abundanceatom or for a single isotope in Abund is ignored.Sys.Nucs = 'Cu,(32,33)S,1H'; % natural Cuwith enriched 33S and a pure 1HSys.Abund = {1,[0.1,0.9],1}; % one waySys.Abund = {[],[0.1,0.9],[]}; % another way,yieldig the same resultAbund Specify abundances for custom isotope mixtures. See Nucs above.n Vector of number of equivalent nuclei, e.g.Sys.Nucs = '1H,13C'; % one class of 1H andone class of 13CSys.n = [2 3]; % 2 protons and 3carbon-13 spinsSys.n can be omitted if all nuclei in Sys.Nucs occur only once.The g valuesThe g matrices/tensors for all electron spins in the system are supplied in two fields:∙g, for the principal values of the g tensors.∙gpa, for the Euler angles specifying the orientations of the g tensors relative to the molecular frame.See also the reference page on the Electron Zeeman interaction.g Depending on whether the g tensors are rhombic, axial orisotropic, different ways of input are possible:∙Rhombic: Each row contains the three principal values of the g tensor of one electron spin.∙Sys.g = [2 2.05 2.3]; %rhombic g, for one electron spin∙Sys.g = [2 2.1 2.3; 1.9 1.95 2.01]; %rhombic g, for two electron spins∙Axial: For axial g tensors, only two values are needed. The first value is the one perpendicular to the easy axis, andthe second is the one parallel to it.∙Sys.g = [2.25 2.03]; %axial g, for one electron spin∙ % =[2.25 2.25 2.03]∙Sys.g = [2.25 2.03; 1.99 1.98]; %axial g, for two electron spins∙ % =[2.25 2.25 2.03; 1.99 1.99 1.98]∙Isotropic: If g is isotropic, it is sufficient to give its isotropic value once.∙Sys.g = 2.005; %isotropic g, for one electron spin∙Sys.g = [2.0023; 2.0025]; %isotropic g, for two electron spins∙No value: If no values are given, EasySpin assumes isotropic g values of 2.0023... (see gfree) for allelectron spins.When the principal values of g are given in g, the orientation of the g tensor can be specified by Euler angles in gpa, see below.As an alternative, it is also possible to give the full g tensor in g, e.g.Sys.g =[ 2.0033971 -0.0004307-0.0000036;...-0.0004145 2.0030324-0.0000063;...-0.0000144 0.0000129 2.0022372];In this case, the Euler angles in gpa are ignored.gpa Each row of this array contains the three Euler angles (inradians) for the passive rotation which transforms the g matrixof the associated electron spin from its eigenframe to themolecular frame (see also the function erot). If not specified,gpa is assumed to be all-zero, that is, all tensors are alignedwith the molecular frame.Sys.gpa = [0 10 0]*pi/180; % oneelectron spinSys.gpa = [0 0 0; 0 pi/4 0; 0 -pi/4 0]; % threeelectron spinsWith these angles, EasySpin can transform a g tensor from itsdiagonal eigenframe representation to the molecular framerepresentation. Here is an explicit example how this is doneinternally:gpa = [10 34 -2]*pi/180; % Euler angles, inradiansg = [2.1 1.97 2.04]; % principal valuesR = erot(gpa); % rotation matrixgdiag = diag(g) % g in eigenframegM = R*gdiag*R.' % g in molecular framewhich gives the following outputgdiag =2.1000 0 00 1.9700 00 0 2.0400gM =2.0797 -0.0147 0.0264-0.0147 1.9728 -0.01150.0264 -0.0115 2.0575Hyperfine couplingsFor each electron-nucleus pair, a hyperfine coupling tensor can be specified. The following fields are used.A Principal values of the hyperfine interaction matrices A/h, inMHz. Each row contains the principal values of the A tensors of one nucleus. A(k,:) refers to nuclear spin k. Orientations of the A matrices are given in Apa.Sys.A = [-6 12 23]; % 1 electron and1 nucleusSys.A = [10 10 -20; 30 40 50]; % 1 electron and2 nucleiFor axial and isotropic hyperfine tensors, the notation can be shortened, just as in the case of the g tensor.Sys.A = [4 10]; % = [4 4 10] (axial, 1electron and 1 nucleus)Sys.A = 34; % = [34 34 34] (isotropic,1 electron and 1 nucleus)Sys.A = [4 10; 1 2]; % = [4 4 10; 1 1 2] (axial,1 electron and2 nucleui)Sys.A = [7; 3]; % = [7 7 7; 3 3 3](isotropic, 1 electron and 2 nuclei)If the system contains more than one electron spin, each row contains the principal values of the hyperfine couplings to all electron spins, listed one after the other.Sys.A = [10 10 -20 30 40 50]; % 2electrons and 1 nucleusSys.A = [10 10 -20 30 40 50; 1 1 -2 3 4 5]; %2 electrons and 2 nucleiIt is possible to specify full A matrices in A. The 3x3 matriceshave to be combined like the 1x3 vectors used when only principal values are defined: For different electrons, put the 3x3 matrices side by side, for different nuclei, on top of each other. If fullmatrices are given in A, Apa is ignored.Sys.A = [5 0 0; 0 5 0; 0 05] % 1 electron and 1nucleusSys.A = [[5 0 0; 0 5 0; 0 0 5]; [10 0 0; 0 100; 0 0 10]] % 1 electron and 2 nucleiSys.A = [[5 0 0; 0 5 0; 0 0 5], [10 0 0; 0 100; 0 0 10]] % 2 electrons and 1 nucleus A_ Spherical form of the hyperfine matrix, in terms of its isotropic,axial and rhombic components. The units are MHz. A and A_cannotbe used at the same time.Sys.A_ = 2; % isotropic component onlySys.A_ = [2 3] % isotropic and axialcomponentSys.A_ = [2 3 0.4] % isotropic, axial andrhombic componentThe cartesian form (as used in A) and the spherical form (as used in A_) are related byA = A_(1) + [-1,-1,2]*A_(2) + [-1,+1,0]*A_(3);For more than one nucleus and more than one electron spin, the array structure isanalagous to A.Apa Array of Euler angles giving the orientations of the various Amatrices in the molecular frame, as described above for gpa. IfApa is not specified, it is assumed to be all-zero, that is, alltensors are aligned with the molecular frame. See also erot. Apahas a layout analogous to A. Each row contains the three Eulerangles for one nucleus.Sys.Apa = [0 20 0]*pi/180; % one electron spin,one nucleusSys.Apa = [0 0 0; 0 10 90; 12 -30 34]*pi/180 %one electron spin, three nucleiIf there are two or more electron spins, each nucleus has two or more hyperfinetensors, and consequently each row should contain two or more sets of Euler angles.Sys.Apa = [0 20 0, 13 -3080]*pi/180; % 2 electrons, 1nucleusSys.Apa = [0 20 0, 0 0 0; 0 10 0, 0 3050]*pi/180; % 2 electrons, 2 nucleiNuclear quadrupole couplingsFor each nucleus spin, a nuclear quadrupole coupling tensor Q can be given, using the fields Q(for e2qQ/h and eta, the principal values, or the fullmatrix) and Qpa (for the orientation).Q Array containing the quadrupole tensors Q/h for all nuclei, in MHz. There are several possible ways to specify the tensors.∙For axial Q tensors: Specify e2Qq/h for each nucleus∙Sys.Q = 0.7 % one nucleus,eeQq/h = 0.7 MHz∙Sys.Q = [0.7, 1.2] % same for twonuclei∙For rhombic Q tensors: Specify e2Qq/h and eta for each nucleus, two numbers per row.∙Sys.Q = [1.2 0.29] % onenucleus, eeQq/h = 1.2 MHz and eta = 0.29∙Sys.Q = [1.2 0.29; 0.1 0] % same witha second nucleus∙For all Q tensors: specify three principal values of the Q tensor, for onenucleus per row.∙Sys.Q = [-1 -1 2] % onenucleus, Qxx = -1, Qyy = -1, Qzz = +2 MHz∙Sys.Q = [-1 -1 2; 0.2 0.3 -0.5] % samefor two nuclei∙You can also give the full 3x3 Q tensor for each nucleus.∙Sys.Q = [-1 0 0; 0 -1 0; 0 0 2]*0.1; %for one nucleus, diagonal∙Sys.Q = [0.125 -0.6495 -1.299;-0.6495 -0.625 0.75; -1.299 0.75 0.5]; %general, one nucleus∙Q1 = [-0.9 0 0; 0 -1.1 0; 0 0 2]*0.15;∙Q2 = [-0.7 0 0; 0 -1.3 0; 0 0 2]*0.31;∙Sys.Q = [Q1; Q2]; % for two nuclei Here is an example of how to convert from e2Qq/h and eta to the three principal Qvalues:eeQq = 1; eta = 0.2;I = 1; % nuclear spin must be known!Q = eeQq/(4*I*(2*I-1)) * [-1+eta, -1-eta, 2]See also the reference page on the nuclear quadrupoleinteraction.Qpa Euler angles alpha, beta and gamma, describing the orientationof the Q tensors in the molecular frame, analogous to gpa andApa. There must be one row of three angles for each nucleus. Theangles are in units of radians, not degrees.Sys.Qpa = [0 pi/4 0]; % onenucleusSys.Qpa = [30 45 0; 10 -30 0]*pi/180; % twonucleiFor axial Q tensors, the first value (for the angle alpha) isirrelevant.Here is how principal values and angles can be converted to a full matrix:Qpv = [-0.9 -1.1 2]*0.125; % principal valuesQpa = [10 45 -30]*pi/180; % Euler anglesR = erot(Qpa); % rotation matrixQ = R*diag(Qpv)*R'; % full matrixZero-field splittingsFor each electron spin, a zero-field splitting can be specified in the fields D and Dpa. See also the reference page on the zero-field interaction.DD gives the zero-field splitting tensors for the electron spinsin the spin system. It should be in units of MHz (1 cm-1 = 29979MHz). It can be specified in several different ways.∙Axial: If the zero-field splitting tensor is axial, give the D value for each electron spin. The orientation of thetensor can be specified in the field Dpa (see below).∙Sys.D = [200]; % 1 electronspin, D = 200 MHz∙Sys.D = [200 -340]; % 2 electronspins, D value each, in MHz∙Sys.D = [200 -340 1100]; % 3 electronspins, D value each, in MHz∙Rhombic: For a zero-field splitting tensor with rhombicasymmetry, give both the D and the E value for each electronspin. The orientation of the tensor can be specified in thefield Dpa (see below).∙Sys.D = [200 10]; % 1electron spin, D = 200 MHz and E = 10 MHz ∙Sys.D = [200 10; 340 90]; % 2electron spins, D and E value for each spin ∙Principal values: For both axial and rhombic symmetry, you can give the three principal values of the D tensor. Theorientation of the tensor can be specified in the field Dpa(see below).∙Sys.D = [-100 -100200]; % 1 electron spin, Dx =Dy = -100 MHz, Dz = 200 MHz∙Sys.D = [-150 -50 200; 450 350-800]; % 2 electron spinsHere is how the principal values can be computed from D and ED = 120;E = 15; % D andE parameters, in MHzSys.D = [-1,-1,2]/3*D + [1,-1,0]*E %conversion to D principal valuesIt is possible to provide a non-traceless D tensor.∙Full matrix: You can also specify the full D matrix for each electron spin. E.g. for one electron spin∙Sys.D = [-33.8 -24.1 -122.1; -24.1-91.2 44.4; -122.1 44.4 125];It is possible to provide a non-traceless D tensor.Include zeros for any electron spin with S = 1/2.Dpa nx3 matrix of realEuler angles describing the orientation of the D tensors,completely analogous to gpa and Qpa. If absent, it is assumedto be all zeros.Internally, EasySpin uses the following procedure to compute the full D tensor in the molecular frame from the given principalvalues and the Euler angles Dpv = [-25 -55 80]; % sample principalvalues Dpa = [10 20 0]*pi/180; % sample tilt anglesR = erot(Dpa) % rotation matrixD = R*diag(Dpv)*R.' % computation of thefull D tensorHigh-order zero-field splittingsEasySpin supports a series of high-order electron spin operators in the spin Hamiltonian. These, howevere, are available only for the first electron spin in a spin system. aF 1x2 array of realValues, in MHz, of the fourth-order parameters a andF. For their definition, see the reference page onhigh-order operators .Sys.aF = [10 -3]; % in MHzBoth a and F are defined in terms the molecularreference frame, which is assumed to coincide thefour-fold symmetry axes of the cubic part. For otherorientations (e.g. along the trigonal axes), thehigh-order parameters B40 and B44 can be used.B20, B21, B22, B40, B41, ...,B44, B60, ...,B66 real or 1x2 arrayThe coefficients for the extended Stevens operators of the electron spin (see also the function stev ).B20, B40 and B60 are scalars. The other fieldsspecify two parameters each. E.g. B43 represents. If only one element is given, it is takenas .Currently, it is not possible to include tilt angles for the principal frames of these high-order interactions. All of them are assumed to be collinear with the molecular frame. By changing the molecular frame, i.e. by tilting all other tensors (g, A, etc), this limitation can be circumvented to some degree. Still, all the high-order interactions are collinear.Electron-electron couplingsFor each electron spin pair, a bilinear coupling tensor (combining isotropic, anisotropic, and antisymmetric exchange) as well as an isotropic biquadratic exchange coupling can be given. See also the reference page on the electron-electron interaction.ee 1xN or Nx3 or 3Nx3 array of realPrincipal value of the electron-electron interaction matrices.Each row corresponds to the diagonal of an interaction matrix(in its eigenframe). They are lexicographically orderedaccording to the indices of the electrons involved. If n is thenumber of electrons, there are N = n(n-1)/2 rows. E.g. for fourelectrons there have to be six rows with the principal valuesfor the interaction of electrons 1-2, 1-3, 1-4, 2-3, 2-4, and3-4, in this order. For two electrons, ee contains one row only.E.g. for two S=1/2, the electron-electron interaction withisotropic and dipolar coupling is Sys.ee = Jiso + Jdip*[11 -2].Sys.S = [1/2 1/2]; % two electronsSys.ee = [50 50 100]; % one coupling matrixSys.S = [1/2 1/2 1/2]; % threeelectronsSys.ee = [50 50 100; 10 10 -30; 0 0 0]; % threecoupling matricesIf only isotropic couplings are needed, one number per couplingis sufficient.Sys.S = [5/2 5/2]; % two electron spinsSys.ee = 10*30e3; % one coupling (1-2)Sys.S = [1/2 1/2 1/2]; % three electronspinsSys.ee = [980 10 10]; % three coupling(1-2,1-3,2-3)Sys.S = [1/2 1/2 1/2 1/2]; % four electronspinsSys.ee = [10 0 0 5 -10 20]; % six couplings(1-2,1-3,1-4,2-3,2-4,3-4)For accomodating antisymmetric exchange, the full 3x3interaction matrices (instead of the 3 principal values and 3Euler angles) can be specified. For a 2-electron system, ee isthen a 3x3 array. For more electrons, the 3x3 matrices arestacked on top of each other.Sys.S = [1/2 1/2]; % twoelectronsSys.ee = [50 0 0;0 50 0; 0 0 100]; % onecoupling matrixSys.S = [1/2 1/2 1/2]; % threeelectronsee12 = [50 0 0; 0 50 0; 0 0 100];ee13 = diag([10 10 -30]);ee23 = zeros(3);Sys.ee = [ee12;ee13;ee23]; %three coupling matriceseepa Nx3 array of realEuler angles describing the orientation of theelectron-electron interaction matrices specified in ee. Each row contains the Euler angles for the corresponding row in ee.ee2 Contains the isotropic biquadratic exchange coupling, in MHz, for each pair of electron spins, The spin pairs are ordered asin ee.Sys.S = [3/2 3/2]; % two electron spinsSys.ee2 = 130; % one biquadraticcoupling 1-2Sys.S = [3/2 3/2 3/2]; % three electron spinsSys.ee = [130 150 190]; % couplings 1-2, 1-3,2-3Sys.S = [1 1 1 1]; % four electron spinsSys.ee = [130 0 130 130 0 130]; % couplings 1-2,1-3, 1-4, 2-3, 2-4, 3-4EasySpin defines the biquadratic exchange term in the spin Hamiltonian as+j(S1.S2)2, with a plus sign. Sometimes, it is defined with a negative sign, so becareful when using literature values.Line broadenings and strainsThere is a number of fields by which line broadenings can be specified. lw and lwEndor are line widths (FWHM) which are used for convolutionof the simulated spectrum. All others are so-called strains and describe Gaussian distributions in the associated spin Hamiltonian parameters.For a full documentation of the line broadening fields in the spin system structure, see the page on line broadenings.Spectral broadeningsOverviewEPR spectra are not infinitely sharp, they are broadened by relaxation, unresolved hyperfine splittings, or distributions in magnetic properties such as g and A values, and others. EasySpin allows you to includebroadening in most spectral simulations (solid-state cw EPR with pepper, liquid EPR with garlic, ENDOR with salt).There are two types of broadenings。