Spectral broadenings

EPR spectra do not consist of a discrete set of infinitely sharp lines. Lines are broadened by dynamic effects (relaxation, tumbling, chemical exchange) or static effects (orientational disorder, unresolved hyperfine splittings, distributions in magnetic properties such as g, A, and D values).

EasySpin takes orientational disorder into account explicitly for solid-state spectra. It allows you to include some forms of additional broadening in most spectral simulations (solid-state cw EPR with pepper, liquid EPR with garlic, ENDOR with salt). In addition, any static broadening model can be implemented using explicit averaging.

The two types of broadening models built into EasySpin are:

• Isotropic convolutional broadenings: A convolutional spectral broadening is computed by convolving the final simulated field-swept or frequency-swept stick spectrum with a Gaussian or Lorentzian line shape of a given width. This broadening method is the simplest possible: it is isotropic and is not based on any physical model causing the broadening. It can be used to visually adjust the broadening of a simulated spectrum to match the one of an experimental one. Since this broadening method does not assume a physical reason for the broadening, it is often called "phenomenological". It should only be applied to S=1/2 systems with small g anisotropy. It works OK for high-spin systems with small zero-field splittings such as organic triplets.
• Anisotropic broadenings: Often, the spectral broadening depends on the orientation of the spin center relative to the external magnetic field. Such broadenings are taken into account in the simulation by adding to the simulated spectrum a Gaussian for each resonance line during the simulation. Physical origins for anisotropic broadenings are unresolved hyperfine splittings and so-called strains. A strain is a distribution in a spin hamiltonian parameter due to small structural variations among the paramagnetic centers in the sample. For example, g strain describes a distribution of g principal values.

The broadenings are given in fields of the spin system structure, which contains the spin system and all associated spin Hamiltonian parameters. Not all types of broadenings are supported by all simulation functions.

Broadenings are treated differently in the simulation of slow-motion cw EPR spectra using chili. See the documentation of chili.

All broadenings are understood to be FWHM (full width at half height) or PP (peak-to-peak), independent of the simulation function, the line shape or the detection harmonic. For the conversion to and from peak-to-peak line widths, see the reference page on line shapes.

Use only broadenings of one type at a time.

This type of broadening is applied by convolving a stick spectrum with a single line shape. Therefore, it applies the same line width and line shape to each transition and orientation.

Isotropic convolutional broadening should only be used for S=1/2 systems with very small g anisotropy, and for S>1/2 systems with small zero-field splittings. For systems with substantial g anisotropy, or higher electron spin with substantial zero-field splitting, it will lead to incorrect results. The reason is that in those cases, distributions in g and D are almost always dominant, and a simple convolutional broadening is inappropriate.

The following fields in the spin system structure specify convolutional broadenings.

Sys.lwpp = 10;       % Gaussian broadening
Sys.lwpp = [0 12];   % Lorentzian broadening
Sys.lwpp = [10 12];  % Voigtian broadening (Gaussian + Lorentzian)

To model anisotropic broadenings in solid-state cw EPR spectra, use fields depending on the physical origin of the broadening:

• For unresolved hyperfine splittings, use HStrain.
• For strains, i.e. distributions in spin Hamiltonian parameters: use gStrain, AStrain and/or DStrain.

More than one of these broadenings can be specified simultaneously. The total broadening for a given orientation is the combination of all individual broadenings.

The broadenings resulting from the various strains are computed in an approximate way. For example, for gStrain, the derivative with respect to g of the resonance field of a given transition is computed, and then the magnitude of this derivative is multiplied by the value from gStrain to give the actual line width. A Gaussian with this line width is then added to the spectrum. A similar procedure is used for all other strains.

This approximation, which corresponds to the first term in a Taylor expansion or to first-order perturbation theory, is valid only as long as the strain distribution width is much smaller than the parameter itself, e.g. a gStrain of 0.02 for a g of 2. If the distributions is wider, an explicit loop (see below) should be used, see below.

Sys.HStrain = [10 10 10];        % 10 MHz Gaussian FWHM broadening in all directions
Sys.HStrain = [10 10 50];        % broadening along the molecular z axis larger than along x and y

If the spin system contains only one electron spin, it is possible to specify combined g and A strain or D strain.

You can model any arbitrary form of inhomogeneous broadening. For this, run explicit loops over distributions of spin Hamiltonian parameters, simulate the associated spectra and sum them up (including weights of the distribution function) to obtain an broadened spectrum.

This approach is necessary if the distributions are so broad as to render the built-in perturbational treatment invalid, or if you want to average over non-Gaussian distributions, or correlated distributions, or distributions over parameters other than g, A, and D.

Here is a basic example for a broad Gaussian distribution of D:

% Set up distribution of D
D0 = 1000;                      % center of distribution, MHz
Dfwhm = 500;                    % FWHM of distribution, MHz
D = linspace(0,2*D0,401);       % range of D values, MHz
weights = gaussian(D,D0,Dfwhm); % associated weights

% Spin system
Sys.S = 1;
Sys.lwpp = 1; % mT

% Experiment
Exp.mwFreq = 9.5; % GHz
Exp.Range = 330+ [-1 1]*100; % mT

% Explicit averaging loop over D distribution
spec = 0;
for iD = 1:numel(D)
  Sys.D = D(iD);
  [B,spec_] = pepper(Sys,Exp);
  spec = spec + weights(iD)*spec_;
end

% Plotting
subplot(2,1,1); plot(D,weights); axis tight
subplot(2,1,2); plot(B,spec); axis tight

If more than one parameter is distributed, then nested loops over a multi-dimensional distribution are needed.