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saffron

Calculation of pulse EPR spectra.

Syntax

saffron(Sys,Exp) saffron(Sys,Exp,Opt) y = saffron(...) [x,y] = saffron(...) [x,y,p] = saffron(...) [x1,x2,y,p] = saffron(...)

See also the examples on how to use `saffron`

.

Description

This function calculates pulse EPR (ESEEM and ENDOR) spectra of powder and single crystals.

The output contain the abscissa data in `x`

(time in microseconds or frequency in MHz) and the simulated data in `y`

(time domain trace or ENDOR spectrum). For 2D experiments such as HYSCORE, the two axes are returned in `x1`

and `x2`

. For ESEEM simulations, `p`

contains the frequency abscissa (in MHz) in `p.f`

and the spectrum obtained by Fourier transform from the time domain data in `p.fd`

.

If you don't request any output, `saffron`

plots the simulated data.

The three input arguments to the function are

`Sys`

: spin system (paramagnetic molecule)`Exp`

: experimental parameters`Opt`

: simulation options

`Sys`

is a spin system structure. Fields available in `Sys`

include all needed for the construction of the spin Hamiltonian. Line broadening parameters used by other simulation functions (`lw`

, `lwpp`

, `gStrain`

, etc.) are not recognized, except `HStrain`

. `HStrain`

is used in excitation window computations (see `Exp.ExciteWidth`

) when orientation selection is wanted.

`saffron`

supports any spin system with one electron spin (arbitrary S) and any number of nuclei.

If no orientation selection is required, then even the g tensor (and the microwave frequency) can be omitted. Only the nuclear parameters (and the field) need to be given:

Sys.Nucs = '14N'; Sys.A_ = [0.2 0.3]; Sys.Q = [-1 -1 2]*0.1;

You can provide the transverse and longitudinal relaxation times in the spin system structure:

`T1T2`

An array

`[T1 T2]`

with two numbers, the longitudinal relaxation time constant T1 and the transverse relaxation time constant T2.
`Exp`

contains the experimental parameters, most importantly the magnetic field and the pulse sequence.

`Field`

Magnetic field (in mT) at which the experiment is performed.

`Sequence`

Specifies the pulse sequence,

`'2pESEEM'`

, `'3pESEEM'`

, `'4pESEEM'`

, `'HYSCORE'`

,
or `'MimsENDOR'`

.
`nPoints`

Number of points in the ESEEM time trace or in the ENDOR spectrum. If only one number is given and a two-dimensional experiment is simulated, the number is used in both dimensions. If you give two numbers, each refers to one dimension. E.g.

`[31 256]`

indicates 31 points in the first dimension and 256 points in the second dimension. If not given, default values are used.
`dt`

For ESEEM, time increment (dwell time), in microseconds. For 1D experiments, this should be a single number. For 2D sequences,

`dt`

applies to both dimensions if it's only one number, alternatively one number for each dimension can be given, e.g. `dt = [0.024 0.008]`

.
`tau`

For three-pulse ESEEM, four-pulse ESEEM, HYSCORE and Mims ENDOR, this gives the value of τ, in microseconds. For two-pulse ESEEM, it is the starting τ value.

`T`

For three-pulse ESEEM and four-pulse ESEEM, this gives the starting T value, in microseconds. For the other sequences, this value has no effect.

`t1,t2`

For HYSCORE, these give the starting values of t_{1} and t_{2}, in microseconds.

`Range`

Contains lower and upper limit of the frequency range for ENDOR. Values should be in MHz, e.g.

`Exp.Range=[0 30]`

.
For orientation selection, the following additional parameters are needed.

`mwFreq`

EPR spectrometer frequency in GHz. Needs only to be given if orientation selection is wanted.

`ExciteWidth`

The microwave excitation bandwidth in MHz (responsible for orientation selection). The excitation profile is assumed to be Gaussian, and

`ExciteWidth`

is its FWHM. The default is infinity. To obtain the full excitation with for a given orientation, `ExciteWidth`

is combined with `HStrain`

from the spin system structure.
For user-defined pulse experiments, the following fields are

`Flip`

List of pulse flip angles in multiples of 90°, e.g.

`[1 2]`

for two-pulse ESEEM and `[1 1 2 1]`

for HYSCORE.
`tp`

List of pulse lengths, in microseconds. If not given, all pulses are assumed to be infinitely short (ideal). If given,

`saffron`

integrates the signal over a small offset distribution. See `Opt.nOffsets`

and `Opt.lwOffset`

. Some of the values in `tp`

can be zero, in which case ideal pulses are used. E.g. `[0.200 0 0]`

is a three-pulse sequence with one selective finite-length pulse followed by two ideal pulses.
`Phase`

List of pulse phases, in multiples of 90°. Specifically, 0 is x, 1 is y, 2 is -x, 3 is -y.

`t`

List of initial delays, in microseconds. E.g.

`[0.2 0.2]`

for two-pulse ESEEM with starting τ of 200 ns, or similarly `[0.2 0.1 0.1 0.2]`

for HYSCORE. The delays are defined to go from the end of one pulse to the beginning of the next (unlike in Bruker spectrometers).
`Inc`

Incrementation information for every inter-pulse delay.

`0`

indicates no incrementation, `1`

increment along first dimension, `2`

increment along second dimensions. Example: `[0 1 2 0]`

for HYSCORE and `[1 1]`

for two-pulse ESEEM.
`Filter`

Coherence filter, one character for each inter-pulse delay.

`'+'`

stands for electron coherence order +1, `'-'`

for -1, `'a'`

for 0 (alpha), `'b'`

for 0 (beta), `'0'`

for 0 (alpha or beta), `1`

for +1 or -1, and `'.'`

for anything. Examples: `'.ab.'`

selects the coherence transfer pathways in HYSCORE that leads to alpha/beta cross peaks.
To simulate single crystals, use

`CrystalOrientation`

An Nx3 array that specifies single-crystal orientations for which the EPR spectrum should be computed. Each row of

`CrystalOrientation`

contains the three Euler rotation angles that transform the crystal frame (C) to the lab frame (L). If `CrystalOrientation`

is empty or not specified, the full powder spectrum is computed.
Exp.CrystalOrientation = [0 0 0]; % single crystal, crystal z axis aligned with B0 Exp.CrystalOrientation = [0 pi/2 0]; % single crystal, crystal z axis perpendicular to B0 Exp.CrystalOrientation = [0 0 0; 0 pi/2 0]; % two crystals Exp.CrystalOrientation = []; % powder

`CrystalSymmetry`

Specifies the symmetry of the crystal. You can give either the number of the space group (between 1 and 230), the symbol of the space group, or the symbol for the point group of the space group (in either Schönflies or Hermann-Mauguin notation).

Exp.CrystalSymmetry = 'P21/c'; % space group symbol Exp.CrystalSymmetry = 11; % space group number (between 1 and 230) Exp.CrystalSymmetry = 'C2h'; % point group, Schönflies notation Exp.CrystalSymmetry = '2/m'; % point group, Hermann-Mauguin notation

When `CrystalSymmetry`

is given, all symmetry-related sites in the crystal are included in the calculation. If `CrystalSymmetry`

is not given, space group 1 (P1, point group C1, one site per unit cell) is assumed.

The fields in the structure `Opt`

specify parameters for the simulation algorithm.

`nKnots`

A number giving the number of orientations (knots) for which spectra are explicitly calculated.

`nKnots`

gives the number of orientations on quarter of a meridian, i.e. between θ = 0 and θ = 90°. The default value is 31, corresponding to a 3° spacing between orientations. For highly anisotropic spectra, esp. for HYSCORE, the value often has to be increased to 181 (0.5° spacing) or beyond.
`TimeDomain`

`0`

(default) or `1`

. Determines whether `saffron`

generates the spectrum by binning all the peaks in the frequency domain or by evolution of all the complex exponentials in the time domain. The frequency-domain binning method is very fast and is therefore used as the default. However, it involves small rounding of peak positions, which can in some cases lead to imperfect phase interferences and small artifacts in the spectrum. The time-domain method is significantly slower, but accurate.
`Expand`

Expansion factor used in the simulation, should be between 0, 1, 2, 3 or 4. The higher, the more accurate is a simulation, but the slower it becomes, especially for 2D simulations. Default values are 4 for 1D and 2 for 2D.

`ProductRule`

Determines whether product rule is used or not (1 or 0). By default, it is not used, but simulations with spin systems with more than two nuclei it might run faster with the product rule. The spectral result is independent of this choice.

`EndorMethod`

Determines which method to use to simulate ENDOR spectra. There are three methods:

`EndorMethod = 1`

: This is the default method. It is a sum-over-transitions method that applies bandwidth-limited rf pi pulses (using single-transition operators in the eigenbasis of the nuclear sub-Hamiltonians) on each nuclear transition in turn. It is able to reproduce inter-nuclear cross suppression effects (implicit triple). All rf pulses are modelled as 180 degree pulses on all allowed nuclear transitions, independent of the nature of the nucleus.`EndorMethod = 2`

: This is an alternative method. It uses a brute-force rf sweep approach: It loops over every frequency on the rf axis and calculates the echo amplitude using the same rf pulse operators as`EndorMethod=1`

.`EndorMethod = -1`

: This is the legacy method (default prior to version 5.0.20). For one nucleus, it is equivalent to`EndorMethod=1`

. For multiple nuclei, it gives wrong results: With`ProductRule=1`

, cross suppression effects are not modelled, and with`ProductRule=0`

, peak positions are wrong.

`nOffsets`

Number of points for the frequency offset integration in the case of finite-length pulses. Typical values are between 10 and 100, but should be determined for each case individually.

`lwOffset`

FWHM (in MHz) of the Gaussian distribution of offset frequencies use in the offset integration in the case of finite-length pulses. Typically, this is about the inverse of the length of the first pulse in a pulse sequence, e.g. 100 MHz for a 10ns pulse.

`logplot`

A

`1`

indicates that the HYSCORE spectrum should be plotted with a logarithmic intensity scale. If `0`

(the default), a linear scale is used.
`Window`

Apodization window used before FFT. See apowin for details.

`ZeroFillFactor`

The factor by which the time domain signal array should be padded with zeros before FFT. E.g. with

`ZeroFillFactor=4`

, a 256-point array is padded to 1024 points.
Algorithm

For both ESEEM and ENDOR, `saffron`

uses matrix-based methods similar to those employed by Mims (1972) to compute frequencies and amplitudes of all peaks. With these peaks, a spectrum histogram is constructed, from which the time-domain signal is obtained by inverse Fourier transform.

For the pre-defined sequences, `saffron`

assumes ideal pulses with standard flip angles (two-pulse ESEEM: 90°-180°, three-pulse ESEEM 90°-90°-90°, HYSCORE 90°-90°-180°-90°).

For systems with several nuclei, `saffron`

by default simulates without using product rules, but can employ them if wanted (see Options).

For high-electron spin systems, all terms in the zero-field splitting, even the nonsecular ones, are taken into account.

To generate the spectrum from the time-domain signal, `saffron`

performs (1) baseline correction, (2) apodization with a Hamming window, (3) zero-filling, and (4) FFT.

All the theory is described in

- Stefan Stoll, R. David Britt

**General and efficient simulation of pulse EPR spectra**

*Phys. Chem. Chem. Phys.*2009, DOI: 10.1039/b907277b

See also