Simulating solid-state cw EPR spectra
Synopsis

This user guide explains how to simulate solid-state cw EPR spectra of powders or single crystals using EasySpin. It is assumed that you are familiar with the basic syntax of Matlab, esp. with structures.

It contains the following topics: There are the following advanced topics:
Invoking the simulation function pepper

Solid-state cw EPR spectra of powders and single crystals are computed by the EasySpin function pepper. It can be called with two or three arguments and returns one, two or three outputs. A few sample calls are

Spec = pepper(Sys,Exp);
[Field,Spec] = pepper(Sys,Exp);
Spec = pepper(Sys,Exp,Opt);
[Field,Spec,Trans] = pepper(Sys,Exp);

Don't forget the ; (semicolon) at the end of the line to suppress output to the screen.

The first argument Sys tells pepper all about the spin system, and the second argument Exp gives the experimental parameters. The third, optional, argument Opt contains settings concerning the simulation itself.

The outputs Field and Spec are the magnetic field and the spectrum, respectively. The optional output Trans lists all transitions that were included in the simulation.

Doing a simulation only requires a few lines of code. A simple example is

Sys = struct('g',[2 2 2.2],'lw',2);
Exp = struct('mwFreq',9.5);
pepper(Sys,Exp);

This simulates and plots the spectrum of an S=1/2 system with an axial g tensor. Copy and paste the code above to your Matlab command window to see the graph.

The spin system

The first input argument to pepper is a structure specifying the spin system. It contains fields for the electron spin(s), the nuclear spins, and the various interaction matrices like g and hyperfine tensor.

For a single unpaired electron spin, the field g (principal values of the g tensor) has to be given. A simple orthorhombic S=1/2 system (e.g., a low-spin Fe3+) is

Sys = struct('g',[1.8 2 2.1]);

pepper automatically assumes S=1/2 for the spin quantum number. For systems with more than one unpaired electron, S has to be specified:

Sys.S = 1;       % for triplet states
Sys.S = 5/2;     % for high-spin Mn2+ or high-spin Fe3+

Nuclear spins are included by specifying Nucs (comma-separated list of nuclei) and A (array of hyperfine tensor principal values, in MHz).

Sys = struct('g',[2 2 2],'Nucs','2H','A',[-1 -1 2]*4.2);

If the A tensor is tilted with respect to the molecular frame, the tilt angles can be provided via the field Apa (standing for "A principal angles")

Sys.Apa = [0 30 0]*pi/180; % [alpha beta gamma] in radians

Similar tilt angle sets can be provided for the g tensor (gpa) and the D tensor (Dpa). The D tensor is specified in the D field.

Sys.D = [-1 -1 2]*50; % MHz

These are the principal values of the D tensor. If only the scalar D and E values are given, they can be converted as follows:

D = 100; E = 10; % in MHz
Sys.D = [-D/3+E, -D/3-E, +2*D/3];

Details about all the spin Hamilton parameters can be found on the spin Hamiltonian reference page. It is also possible to include several electron spins. Refer to the page about spin system structures for details.

Broadenings

No cw EPR spectrum is infinitely sharp. Lines are usually broadened due to several reasons. pepper provides means to include several line broadening models in a simulation.

The simplest way to include line broadening is to convolute a stick spectrum with a (Gaussian or Lorentzian) lineshape after the end of the simulation. Such a convolution broadening is specified in the spin system field lw.

Sys.lw = 0.5; % mT

The line width is in mT and refers to FWHM (full width at half height). By default, pepper uses a Gaussian profile. A Lorentzian can be specified in the options structure field LineShape. For details about line shapes and conversion formulas to/from FWHM and peak-to-peak widths, see the page on line shapes.

Physically, the largest contributions to broadening are due to unresolved hyperfine couplings and to distributions in the various spin Hamiltonian parameters that result from structural variations from one paramagnetic center to the next.

To include effects from unresolved hyperfine couplings, an orientation-dependent phenomenological broadening can be specified in HStrain:

Sys.HStrain = [50 50 87];   % [along x, along y, along z], in MHz

Distributions of g and A are given in similar fields:

Sys.gStrain = [0.01 0.02 0.005];
Sys.AStrain = [10 10 30]; % in MHz

The three values in gStrain are the FWHM parameters of the Gaussian distributions of the respective g principal values given in Sys.g. AStrain is the same for the A tensor.

Distributions of the D tensor values can be given in DStrain, where the first value is the width of the (scalar) D distribution, and the second is the width for the E distribution.

All these broadening parameters can be combined. However, usually a modelling of the broadening with the phenomenological HStrain is absolutely sufficient.

Basic experimental settings

All experimental settings are given in the second input argument Exp. Just as the spin system, Exp is a structure containing several fields.

For every simulation, the spectrometer frequency has to be given in the field mwFreq in units of GHz (Gigahertz).

Exp.mwFreq = 9.754; % in GHz

There are two ways to specify the magnetic field sweep range.

Exp.CenterSweep = [340 80]; % in mT
Exp.Range = [300 380];      % in mT

Either the center and the sweep width (in mT) are given in Exp.CenterSweep, or the lower and upper limit of the sweep range (again in mT) are given in Exp.Range. In many cw EPR spectrometers, the field range is specified using a center field and a sweep width, so Exp.CenterSweep is the more natural choice.

Exp.CenterSweep and Exp.Range are only optional. If both are omitted, garlic automatically chooses a field range large enough to accomodate the full spectrum. If both are given, garlic takes the values given in Exp.CenterSweep and ignores those in Exp.Range.

By default, pepper computes a 1024-point spectrum, just as commerical spectrometers measure by default. However, the number of points can be changed manually to a different value, e.g.,

Exp.nPoints = 5001;

By default, pepper computes the first-derivative spectrum. By changing Exp.Harmonic, the absorption or the second-derivative spectrum can be requested.

Exp.Harmonic = 0; % absorption spectrum
Exp.Harmonic = 1; % first derivative
Exp.Harmonic = 2; % second derivative
More experimental settings

For more advanced spectral simulations, pepper offers more configuration possibilities in the experimental parameter structure Exp.

Most cw EPR resonators operate in perpendicular mode, i.e., the oscillating magnetic field component of the microwave in the resonator is perpendicular to the static field. Some resonators can operate in parallel mode, where the microwave field is parallel to the static one. pepper can simulate the resulting spectra:

Exp.Detection = 'parallel'; % parallel detection

The polarizing effects of low sample temperatures can also be included in the simulation by specifying the temperature:

Exp.Temperature = 4.2; % temperature in Kelvin

With this setting, pepper will include the relevant polarization factors resulting from a thermal equilibrium population of the energy levels.

Powders and crystals

If not specified otherwise, pepper computes powder spectra. But it is as well straightforward to simulate spectra for single crystals. The orientation of the single crystal(s) can be provided in the experiment structure field Exp.Orientations. This field should contain the tilt angles between molecular and laboratory frame (right-handed coordinate system with z along the static field and x along the microwave magnetic field), one set of three angles per column.

For a single paramagnetic center with its molecular frame aligned with the laboratory frame, the setting is

Exp.Orientations = [0;0;0];

In many crystals, there are several sites with identical paramagnetic centers differing only in their orientation relative to the crystal lattice. For such a case, several columns of tilt angle triplets can be specified.

Exp.Orientations(:,1) = [0;0;0];
Exp.Orientations(:,2) = [0;pi/4;0];

or, equivalently,

Exp.Orientations = [0 0 0;0 pi/4 0].';

In this case, pepper returns the sum of the two resulting spectra. To obtain the two spectra separately, Opt.Output has to be set to 'separate' (see below).

If Exp.Orientations set to [] (an empty array), pepper simulates the powder spectrum.

The third input argument to pepper contains simulation options. All of them have reasonable default values, but sometimes it might be necessary to change one of them. In the following the most important ones are presented.

If you want pepper to print information about the simulation to the command window during the computation, use

Options = struct('Verbosity',1);

'Verbosity' specifies the logging level. 0 (the default) suppresses all output, 1 is normal logging, and 2 prints more information, relevant only for debugging.

Another useful option is nKnots, which determines how many orientations pepper will include in the simulation of a powder spectrum. If this value is too low, the spectrum shape contains ripples. nKnots is the number of orientations between the z axis and the x axis (between theta = 0 and theta = 90 degrees).

Options.nKnots = 31; % corresponds to 3-degree increments

The higher nKnots, the finer the orientational grid. The default value of 19 (5-degree increments) is appropriate for most systems. A value larger than 181 (0.5-degree increments) is rarely needed.

After having computed the spectrum for a number of orientations specified by nKnots, the simulation function interpolates these spectra for additional orientations before summing up all spectra. This interpolative refinement can be configured with the field nSpline. nSpline = 4 means that pepper interpolates additional 4 spectra between two adjacent orientations evaluated.

Options.nSpline = 10;  % massive interpolation
Options.nSpline = 0 ;  % switches interpolation off

The option Output can be used to determine the form in which pepper returns the spectral data.

Options.Output = 'separate'; % single crystal: orientations separately
                             % powders: transitions separately
Options.Output = 'summed'; % returns the sum over all orientations and transitions

There are more option fields available. For details, see the documentation page on pepper.

Simulations of spectra from systems with more than one nucleus can be very time-consuming. To accelerate such computations, nuclei with small hyperfine couplings can be treated by first-order perturbation theory.

E.g., in the simulation of the Cu porphyrin system

Sys = struct('S',1/2,'g',[2 2 2.2]);
Sys = nucspinadd(Sys,'63Cu',[50 50 500]);
A = [20 20 30];
Sys = nucspinadd(Sys,'14N',A);
Sys = nucspinadd(Sys,'14N',A);
Sys = nucspinadd(Sys,'14N',A);
Sys = nucspinadd(Sys,'14N',A);
Sys.lw = 0.6;

large 648x648 matrices are involved, and the simulation needs several hours. Using the perturbational treatment of the 4 nitrogens, the matrix size can be reduced to 8x8, and the simulation completes within a couple of seconds.

To enable the perturbational treatment of super-hyperfine (SHF) nuclei, set the field Perturb in the options structure to 1

Exp = struct('mwFreq',9.5,'Range',[270 360]);
Opt = struct('nKnots',31);
Opt.Perturb = 1;
[x,y] = pepper(Sys,Exp,Opt);

pepper will automatically determine those nuclei where first-order perturbation theory can be applied without introducing noticeable errors in the simulated spectrum.

If you want to compare the perturbational treatment to the full treatment, set Perturb to 0 and Threshold to 0.

Opt.Perturb = 0;
Opt.Threshold = 0; % important!

The second line tells pepper to include all possible transitions in the simulation. This is important, since due to many anticrossings between nuclear sublevels the usual transition pre-selection will miss transitions, possibly leading to distorted spectra.

pepper can handle both thermal equilibrium and non-equilibrium populations. Both are specified in the field Temperature of the experimental settings structure.

For thermal equilibrium, just give the temperature in Kelvin:

Exp.Temperature = 77; % 77K, boiling point of liquid nitrogen

For non-equilibirum populations, Temperature must be a vector. If the spin systems contains N electron states, then this vector must contain N elements, each specifying the population of one of the electron states at zero field, sorted according to their energy from lowest to highest.

E.g., an organic triplet with S=1 and I=1 has 3 electron states, each further split into three sublevels by the coupling to the nuclear spin. The population vector in this case should contain three elements:

Exp.Temperature = [0.6 0.8 1.1]; % highest state is most populated

This specifies that all the sublevels of the lowest zero-field electron states have a population of 0.6, etc. The sublevels of the highest-energy zero-field electron state have a population of 1.1. The populations don't have to be normalized, pepper takes care about that.

To compute the state populations for a non-zero field state, pepper decomposes it into a linear combination of zero-field states and combines the zero-field populations using the resulting linear combination coefficients.

A simple example of a non-equilibrium triplet system is

Sys = struct('S',1,'g',[2 2 2],'D',[-1 -1 2]*100,'lw',0.2);
Exp = struct('mwFreq',9.5,'Range',[320 360],'Harmonic',0);
Exp.Temperature = [0.5 0.6 0.9];
[x,y] = pepper(Sys,Exp);
plot(x,y);