saffron
Synopsis

Calculation of pulse EPR spectra. 

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 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;

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', '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. The defaults for nPoints are 512 for 1D and 256 for 2D experiments.
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, 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, 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 t1 and t2, in microseconds.
T1T2 An array [T1 T2] with two numbers, the longitudinal relaxation time constant T1 and the transverse relaxation time constant T2.
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 excitation width of the microwave 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 1 2 1] for HYSCORE.
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. Currently, not all incrementation schemes are supported.
t List of initial delays, in microseconds.
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 pulse followed by two ideal hard pulses.
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

Orientations 3xN array of real
Specifies single-crystal orientations for which the ENDOR spectrum should be computed. Each column of Orientation contains the three Euler rotation angles [phi;theta;chi] of the magnetic field in a molecule fixed frame. If Orientation is empty or not specified, the full powder spectrum is computed. 
Exp.Orientations = [0;0;0];              % crystal with z axis aligned with B0
Exp.Orientations = [0;pi/2;0];           % crystal with z axis perpendicular to B0
Exp.Orientations = [0 0 0;0 pi/2 0].';   % two orientations
Exp.Orientations = [];                   % powder
CrystalSymmetry

Specifies the symmetry of the crystal, if single-crystal spectra to be simulated (that is, if Exp.Orientations is specified). The crystal symmetry can be either the number of the space group (between 1 and 230), the symbol of the space group (e.g. 'P21212' or the symbol for the point group (e.g. 'C2h' or '2/m').

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.
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.
nOffsets Number of points for the 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

See also

nucfrq2d, salt, pepper