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Overview

EPR 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 include broadening in most spectral simulations (solid-state cw EPR with pepper, liquid EPR with garlic, ENDOR with salt).

There are two types of broadenings

- Isotropic convolutional broadenings: A convolutional spectral broadening is computed by convoluting the final simulated stick spectrum with a Gaussian or Lorentzian line shape of a given width. This broadening method is the simplest possible: it is isotropic and does not assume 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.
- Anisotropic broadenings: Often, the spectral broadening depends on the orientation of the spin system 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. E.g., 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.

Warning! This type of broadening should only be used for S=1/2 systems with very small g anisotropy. For systems with higher electron spin or with substantial g anisotropy, it will lead to incorrect results.

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

`lwpp`

Line width for isotropic broadening (PP, peak-to-peak), used for convolution of a field-swept or frequency-swept liquid or solid-state cw EPR spectrum. Peak-to-peak refers to the horizontal distance between the maximum and the minimum of a first-derivative lineshape.

For field-swept spectra, the unit of `lwpp`

is mT. For frequency-swept spectra, the unit is MHz.

- 1 element:
`Gaussian`

- 2 elements:
`[Gaussian Lorentzian]`

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

For conversion between FWHM and PP line widths, see the reference page on line shapes.

`lw`

Same as

`lwpp`

, except that the numbers are assumed to indicate the full width at half maximum (FWHM) instead of the peak-to-peak (PP) width. For conversion between FWHM and PP line widths, see the reference page on line shapes.
`lwEndor`

Line width (FWHM) for ENDOR broadening. Usage is the same as

`lw`

. For `lwEndor`

, no peak-to-peak analogue is available. See the page on line shapes for conversion formulas.
Anisotropic broadenings in solid-state cw EPR spectra has two main physical origins:

- unresolved hyperfine couplings (
`HStrain`

) - strains, i.e. distributions in Hamiltonian parameters (
`gStrain`

,`AStrain`

and`DStrain`

).

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

`HStrain`

Residual line width (full width at half height, FWHM), in MHz, describing broadening due to unresolved hyperfine couplings. The three components are the Gaussian line widths in the x, y and z direction of the molecular frame.

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

The line width for a given orientation
of the static
magnetic field is given by
where , and
are the three elements of `HStrain`

.

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

`gStrain`

`[FWHM_gx FWHM_gy FWHM_gz]`

Defines the g strain for the electron spin. It specifies the FWHM widths of the Gaussian distributions of the g principal values (x, y and z). The distributions are assumed to be completely uncorrelated. g strain is not supported for systems with more than one electron spin.

`AStrain`

`[FWHM_Ax FWHM_Ay FWHM_Az]`

, in MHz
Vector of FWHM widths (in Megahertz) of the Gaussian distributions of the corresponding principal values in

`A`

(x, y, z) of the first nucleus in the spin system. The distributions are completely uncorrelated. AStrain is not supported for systems with more than one electron spin.
To set the correlation between `gStrain`

and `AStrain`

, use the field `gAStrainCorr`

.
`gAStrainCorr`

`+1`

(default) or `-1`

Sets the correlation between

`gStrain`

and `AStrain`

. `+1`

indicates that positive change in gx is correlated with a positive change in Ax etc,whereas `-1`

indicates anticorrelation.
`DStrain`

`FWHM_D`

or `[FWHM_D FWHM_E]`

Widths (FWHM) in MHz of the Gaussian distributions of the scalar parameters D and E that specify the D matrix of the zero-field interaction. If

`FWHM_E`

is omitted, it defaults to zero.
If the spin system contains more than one electron spin, `DStrain`

should contain one row for each electron spin.

Examples:
`DStrain = [10, 5]`

specifies a Gaussian distribution of D with a FWHM of 10 MHz and a Gaussian distribution of E with a FWHM of 5 MHz. `DStrain = [100 33]`

specifies a Gaussian distribution of D with a FWHM of 100 MHz and a Gaussian distribution of E with a FWHM of 33 MHz. For two electron spins, `DStrain = [10 5; 100 20]`

specifies `[10 5]`

for the first electron spin and `[100 20]`

for the second.

`DStrainCorr`

`+1`

(default) or `-1`

Sets the correlation coefficient between

`D`

and `E`

, which can be between -1 and +1. `+1`

indicates that positive change in D is correlated with a positive change in E, whereas `-1`

indicates anticorrelation. If not given, it defaults to 0, and the distributions in D and E are uncorrelated.
If the spin system has more than one electron spin, give one correlation coefficient per electron spin.

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.

Using your own broadening models

When none of the above inhomogeneous broadenings apply to your problem, you can always run a loop over any distribution of spin Hamiltonian parameters, simulate the associated spectra and sum them up (including weights of the distribution function) to obtain an inhomogeneously broadened line.