Simulating slow-motion cw EPR spectra

This user guide explains how to simulate slow-motion cw EPR spectra of using EasySpin's function chili. It is assumed that you are familiar with the basics of MATLAB, esp. with structures.

This user guide contains the following topics: There are the following advanced topics:
Running the simulation

Slow-motion cw EPR spectra of S=1/2 systems are computed by the EasySpin function chili.

chili(Sys,Exp)

It is called with two arguments. The first argument Sys tells chili all about the static and dynamic parameters of the spin system. The second argument Exp gives the experimental parameters.

If no output argument is given, chili plots the computed spectrum. But it can also return one or two outputs. (Don't forget the semicolon at the end of the line to suppress output to the command window.)

Spec = chili(Sys,Exp);
[Field,Spec] = chili(Sys,Exp);

The outputs Field and Spec are vectors containing the magnetic field axis and the spectrum, respectively. If these are requested, chili does not plot the spectrum.

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

Sys = struct('g',[2.008,2.006,2.003],'Nucs','14N','A',[20,20,85]);
Sys.tcorr = 4e-8;
Exp = struct('mwFreq',9.5);
chili(Sys,Exp);

The first line defines the spin system, a nitroxide radical with anisotropic g and A tensors. The second line gives the rotational correlation time of the radical. The third line specifies experimental parameters, here only the microwave frequency (The magnetic field range is chosen automatically). The fourth line calls the simulation function, which plots the result. Copy and paste the code above to your MATLAB command window to see the graph.

Of course, the names of the input and output variables don't have to be Sys, Exp, Field and Spec. You can give them any name you like, as long as it is a valid MATLAB variable name, e.g., FremySaltSolution or QbandExperiment. For convenience, thoughout this tutorial, we will use short names like Sys and Exp.

Specifying the static parameters

The first input argument specifies the static parameters of the paramagnetic molecule. It is a MATLAB structure with various fields giving values for the spin system parameters.

Sys.g = [2.008,2.006,2.003];
Sys.Nucs = '14N';
Sys.A = [20,20,80];  % MHz

The first line defines the g tensor of the spin system via its three principal values. chili always assumes a single unpaired electron spin S=1/2.

The field Sys.Nucs contains a string giving all the magnetic nuclei in the spin system, a nitrogen-14 in the above example. Use a comma-separated list of isotope labels to give more than one nucleus. E.g., Sys.Nucs = '14N,1H,1H' specifies one nitrogen and two different protons. See the section on multinuclear systems about details of the slow-motion simulation in that case.

Sys.A gives the hyperfine couplings in MHz (Megahertz), with three principal values on a row for each of the nuclei listed in Sys.Nucs. The following defines a hydrogen atom with a 10 MHz coupling to the unpaired electron and a 13C atom with a 12 MHz coupling.

Sys.Nucs = '1H,13C';
Sys.A = [10 12]; % MHz

Remember that chili (and other EasySpin functions, too), take the hyperfine coupling values to be in MHz. Often, values for hyperfine couplings are given in G (Gauss) or mT (Milltesla), so you have to convert these values. For g = 2.00232, 1 G corresponds to 2.8025 MHz, and 1 mT corresponds to 28.025 MHz. The simplest way to convert coupling constants from magnetic field units to MHz is to use the EasySpin function mt2mhz:

A_MHz = mt2mhz(A_mT);    % mT -> MHz conversion
A_MHz = mt2mhz(A_G/10);  %  G -> MHz conversion (1 G = 0.1 mT)

The orientations of the tensors relative to the molecular frame are defined in terms of Euler angles in 3-element array (see the function erot).

Sys.gFrame = [0 0 0];    % Euler angles for g tensor
Sys.AFrame = [0 pi/4 0]; % Euler angles for A tensor

For more details on these static parameters, see the documentation on spin systems.

Dynamic parameters

The spin system structure also collects parameters relating to the dynamics of the paramagnetic molecules.

There are several alternative ways to specify the rate of isotropic rotational diffusion: either by specifying tcorr, the rotational correlation time in seconds

Sys.tcorr = 1e-7;   % 10^-7 s = 100 ns

or by givings its base-10 logarithm

Sys.logtcorr = -7;   % 10^-7 s = 100 ns

or by specifying the principal value of the rotational diffusion tensor (in s-1)

Sys.Diff = 1e9;  % 1e9 s^-1 = 1 ns^-1

or by givings its base-10 logarithm

Sys.logDiff = 9;   % 1e9 s^-1 = 1 ns^-1
Diff and tcorr are related by
Diff = 1/6/tcorr;
The input field Diff can be used to specify an axial rotational diffusion tensor, by giving a 2-element vector with first the perpendicular and the the parallel principal value:
Sys.Diff = [1 2]*1e8;  % in hertz

The lw field has the same meaning As for the other simulation functions garlic and pepper, the field Sys.lw can be used to specify a Gaussian and a Lorentzian broadening (FWHM, in mT)

Sys.lw = [0.05 0.01];   % [GaussianFWHM, LorentzianFWHM] in mT

For the reliability of the simulation algorithm we recommend to always use a small residual Lorentzian line width in Sys.lw

chili is also capable of simulating spectra including Heisenberg spin exchange. The effective exchange frequency (in MHz) is specified in Sys.Exchange, e.g.
Sys.Exchange = 100;     % 100 MHz

A short example of a nitroxide radical EPR spectrum with exchange narrowing is

Nx = struct('g',[2.0088, 2.0061, 2.0027],'Nucs','14N','A',[16 16 86]);
Nx.tcorr = 1e-7; Nx.lw = [0 0.1]; Nx.Exchange = 100;
Exp = struct('mwFreq',9.5,'CenterSweep',[338, 10]);
chili(Nx,Exp);
The orienting potential

chili can also include an orienting potential in the simulation. It is specified in the field lambda in the spin system structure. Up to five coefficients can be given, λ2,0, λ2,2, λ4,0, λ4,2, and λ4,4, in that order. For example,

Nx.lambda = [0.3, -0.2];

indicates λ2,0 = 0.3 and λ2,2 = -0.2.

Basic experimental settings

The second input argument, Exp, collects all experimental settings. Just as the spin system, Exp is a structure containing several fields.

Microwave frequency. To simulate an EPR spectrum, EasySpin needs at a minimum the spectrometer frequency. Put it into Exp.mwFreq, in units of GHz.

Exp.mwFreq = 9.385;  % X-band
Exp.mwFreq = 34.9;   % Q-band

Field range. There are two ways to enter the magnetic field sweep range. Either give the center field and the sweep width (in mT) in Exp.CenterSweep, or specify the lower and upper limit of the sweep range (again in mT) in Exp.Range.

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

On many cw EPR spectrometers, the field range is specified using center field and sweep width, so Exp.CenterSweep is often the more natural choice.

Exp.CenterSweep and Exp.Range are only optional. If both are omitted, EasySpin tries to determine a field range large enough to accomodate the full spectrum. This automatic ranging works for most common systems, but fails in some complicated situations. EasySpin will issue an error when it fails.

Points. By default, pepper computes a 1024-point spectrum. However, you can change the number of points to a different value using

Exp.nPoints = 5001;

You can set any value, unlike some EPR spectrometers, where often only powers of 2 are available (1024, 2048, 4096, 8192).

Harmonic. By default, EasySpin computes the first-harmonic absorption spectrum, i.e. the first derivative of the absorption spectrum. By changing Exp.Harmonic, you can request the absorption spectrum directly or the second-harmonic (second derivative) of it.

Exp.Harmonic = 0; % absorption spectrum, direct detection
Exp.Harmonic = 1; % first harmonic (default)
Exp.Harmonic = 2; % second harmonic

Modulation amplitude. If you want to include effects of field modulation like overmodulation, use Exp.ModAmp

Exp.ModAmp = 0.2; % 0.2 mT (2 G) modulation amplitude, peak-to-peak

Time constant. To include the effect of the time constant, apply the function rcfilt to the simulated spectrum.

More experimental settings

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

Mode. 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. EasySpin can simulate both types of spectra:

Exp.Mode = 'perpendicular'; % perpendicular mode (default)
Exp.Mode = 'parallel';      % parallel mode

Temperature. 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, EasySpin will include the relevant polarization factors resulting from a thermal equilibrium population of the energy levels. For S=1/2 systems, it is not necessary to include the temperature. However, it is important in high-spin systems with large zero-field splittings, and in coupled spin systems with exchange couplings.

Microwave phase. Occasionally, the EPR absorption signal has a small admixture of the dispersion signal. This happens for example when the microwave phase in the reference arm is not absolutely correctly adjusted. EasySpin can mix dispersion with absorption if a Lorentzian broadening is given:

Sys.lwpp = [0.2 0.01];           % Lorentzian broadening (2nd number) required

Exp.mwPhase = 0;                 % pure absorption
Exp.mwPhase = pi/2;              % pure dispersion
Exp.mwPhase = 3*pi/180;          % 3 degrees dispersion admixed to absorption
Powder vs single orientation

In a frozen solution of spin-labelled protein, the protein molecules assume all possible orientations. For slow-motion spectra, this orientational distribution has to be taken into account if a orienting potential is present. If not, it is sufficient to compute only one orientation, as the spectra from all orientations are identical.

The summation of slow-motion spectra over all possible orientations of an immobile protein ("director") is historically called the MOMD (microscopic order macroscopic disorder) model. This is equivalent to a powder spectrum. In chili, the powder spectrum is simulated whenever you specify an ordering potential. To get a single-crystal spectra, i.e. the slow-motion spectrum for the nitroxide attached to a rigid protein with a single orientation, give the crystal orientation in Exp.CrystalOrientation. Exp.CrystalOrientaiton contains the Euler tilt angle (in radians), betwen the lab frame (which is lab-fixed) and the frame of the orienting potential (which is fixed to the protein).

When chili performs a powder simulation, it takes the number of orientations to include from Opt.nKnots. Often, Opt.nKnots does not have to be changed from its default setting, but if the spectrum does not appear to be smooth, Opt.nKnots can be increased. Of course, this also increases the simulation time. For quick simulations, Opt.nKnots should be minimized.

Simulation parameters

The third input structure, Opt, collects parameters related to the simulation algorithm.

The field Verbosity specifies whether chili should print information about its computation into the MATLAB command window. By default, its value is set to 0, so that chili is silent. It can be switched on by giving

Opt.Verbosity = 1;     % log information

Another important option is LLKM. It specifies the number of orientational basis functions used in the simulation. For spectra in the fast and intermediate motion regime, the preset values don't have to be changed. However, close to the rigid limit, the default settings of LLKM might be too small. In that case, LLKM has to be increased, e.g.

Opt.LLKM  = [24 20 10 10];

To see the values of LLKM that chili is using, set Opt.Verbosity=1, as described above.

Systems with more than one nucleus

chili is not capable of simulating a slow-motional cw EPR spectrum of systems with more than one nucleus by solving the Stochastic Liouville equation.

However, if the hyperfine coupling of one nucleus is significantly larger than those of the other nuclei, chili uses an approximate procedure: The slow-motional spectrum simulated using only the electron spin and the nucleus with the dominant hyperfine coupling is convoluted with the isotropic splitting pattern due to all other nuclei. This post-convolution technique gives resonable results.

A simple example is

CuPc = struct('g',[2.0525 2.0525 2.1994],'Nucs','63Cu,14N','n',[1 4]);
CuPc.A = [-54 -54 -608; 52.4 41.2 41.8];
CuPc.logtcorr = -7.35;
Exp = struct('mwFreq',9.878,'CenterSweep',[330 120],'nPoints',5e3);
Opt.LLKM = [36 30 8 8];
chili(CuPc,Exp,Opt);
Frequency sweeps

chili, like the other cw EPR simulation functions pepper and garlic, does field sweeps by default. However, you can use it to simulate frequency-swept spectra as well.

For this, all you need to do is the following

Here is an example of a frequency-swept slow-motion spectrum of a nitroxide radical:

clear
Nx.g = [2.008 2.006 2.002];
Nx.Nucs = '14N';
Nx.A = [20 20 100];
Nx.tcorr = 3e-9;
Exp.Field = 340;         % static field, in mT
Exp.Range = [9.3 9.7];   % frequency range, in GHz
chili(Nx,Exp);

By default, chili returns the absorption spectrum (Exp.Harmonic=0) when you simulate a frequency-swept spectrum. To get the first or second derivative, change Exp.Harmonic to 1 or 2. Note however that Exp.ModAmp is not supported for frequency sweeps.

All other capabilities of chili apply equally to frequency sweep and to field sweeps. For example, you can simulate multi-component spectra, you can use an ordering potential, you can run powder spectra, and you can adjust the basis size.