EasySpin: endorfrq

endorfrq

Synopsis

Computes ENDOR frequencies and transition amplitudes.

Pos = endorfrq(Sys,Par)
Pos = endorfrq(Sys,Par,Opt)
[Pos,Int] = endorfrq(...)
[Pos,Int,Tra] = endorfrq(...)

Description

endorfrq computes ENDOR frequencies and intensities, which can be used to build single-crystal spectra or crystal rotation patterns. endorfrq is used by salt, the solid-state ENDOR spectrum simulation function, to obtain ENDOR positions and intensities. The calling syntax of endorfrq is very similar to that of resfields, its EPR analogue.

The two mandatory input parameters characterise the spin system (Sys), and the experiment parameters (Par) containing the spin system's orientations in the external field. A third parameter Opt can be used to modify and tune endorfrq's computations.

Sys is a spin system structure. In addition to all the fields necessary to construct a spin Hamiltonian, endorfrq only uses the EPR line width field HStrain.

Par is a structure containing information related to the ENDOR experiment.

`mwFreq`	Microwave frequency used in the ENDOR experiment, in GHz.
`Field`	Magnetic field at which the ENDOR spectrum is to be computed, in mT.
`Orientations`	A list of orientations for which resonance fields should be computed. It can be either a 2xn or a 3xn array, containing the orientations along columns. Either two (φ, θ) or three (φ, θ, χ) Euler angles (in radians) characterise each orientation. φ, in the first row, is the angle between the x axis and the xy plan projection of the orientation of the external field in the reference frame of the spin system. θ, in the second row, is the angle at which the external field is off the z axis of the reference frame. The optional χ, in the third row, specifies the third Euler angle and fixes the x axis of the laboratory in the reference frame of the spin system. Altogether, these three angles define the relative orientation between the molecular reference frame and the laboratory coordinate system. Resonance fields depend only on the first two angles, intensities also on the third. If the third angle is not given, EPR intensities are integrated over all possible values of χ.
`Temperature`	Temperature at which the experiment is performed, in K. If omitted (or set to `inf`), no temperature effects are computed.
`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 `inf`.
`CrystalSymmetry`	Specifies the symmetry of the crystal. 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'`). Exp.CrystalSymmetry = 'P21/m'; % 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, `endorfrq` automatically computes the spectra of all symmetry-related sites in the crystal. If `CrystalSymmetry` is not given, `endorfrq` assumes space group 1 (P1, point group C1), which has only one site per unit cell.

Opt contains a set of optional parameters used to adjust the computation to one's needs. If a field or the entire structure is omitted, default values are used. One part of the fields concerns the automatic or manual selection of transitions to include in the ENDOR computation, the other one allows to modify the ENDOR intensity calculation.

`Verbosity`	Level of display. `Opt.Verbosity=0` (default) means that `endorfrq` does not print to the command window. `Opt.Verbosity=1` prints some log messages, higher values are given ever more details.
`Transitions`	mx2 vector of integer Determines manually the level pairs which are used in the spectrum calculation. If given, `endorfrq` uses them and skips its automatic transition selection scheme. Level pairs are specified in `Transitions(k,:)` by the level numbers, starting with 1 for the lowest-energy state.
`Threshold`	Specifies the relative threshold for `endorfrq`'s automatic transition selection scheme. Any transition with a relative average amplitude less than this number is not included in the calculation. The relative average amplitude of the strongest ENDOR transition found is 1. If levl pairs are manually specified in `Transitions`, the threshold setting is ignored.
`Nuclei`	vector Determines which nuclear Zeeman terms should be included in the automatic transition selection procedure. If a system contains two different types of nuclei, this allows the user to tune `endorfrq` to select only transitions belonging to a certain type of nuclei. 1 is the first nucleus in the spin system, 2 the second, and so on. If this field is absent, all nuclei are included by default. E.g. `Opt.Nuclei=2` for a spin system with `Sys.Nucs='63Cu,1H'` will only include ¹H ENDOR transitions.
`Intensity`	`'on'` (default) or `'off'` By default, ENDOR intensities are computed from ENDOR and EPR transition amplitudes between states and the EPR excitation width. If `Intensity` is set to `off`, no intensities are computed. If the `endorfrq` output parameter `Int` is given, it is set to empty. You can also switch off intensity computations by not giving the corresponding output parameter.
`Enhancement`	`'off'` (default) or `'on'` If `on`, `endorfrq` includes the hyperfine enhancement effect in the computation of the transition matrix elemenets by using the full Zeeman part of the spin Hamiltonian (elecron plus nuclear part). Otherwise only the nuclear Zeeman terms are used. The hyperfine enhancement effect causes intensity asymmetries of ENDOR lines at low fields, but it is often compensated by the characteristics of the RF coils used in ENDOR experiments.

There are three output parameters to endorfrq, only the first one is mandatory.

Pos contains the positions of the ENDOR lines (in Megahertz). One column in the output corresponds to one orientation.

Int returns the ENDOR intensities, in an array the same size as Pos. If intensity computations have been switched off, this array is empty. If Int is omitted, intensities are not computed at all.

Tra is a list of the transitions included in the computation. It is a nx2 array containing pairs of level numbers along rows. Levels numbers relate to their energy: the lowest state in energy has number 1, the second lowest is 2, etc. See also the Transitions field in Opt.

Algorithm

endorfrq uses full matrix diagonalization to compute frequencies and amplitudes.

Examples

The following code produces a plot of ENDOR frequencies and amplitudes as a function of the Q tensor axial parameter eeQq. First we define the three input structures to endorfrq and the range of eeQq values.

Sys = struct('S',1/2,'g',2,'Nucs','14N','A',[8 9 10]);
Exp.Field = 350;
Exp.Orientations = [10;40]*pi/180;
Opt.Threshold = 0;
Opt.Enhancement = 'on';

eeQq = 0.0:.1:4;

Next we loop over all eeQq values and compute the associated ENDOR positions and amplitudes for an arbitrary orientation. The results are stored along columns in the array p and w.

for i = 1:numel(eeQq)
  Sys.Q = eeQq(i);
  [p(:,i), w(:,i)] = endorfrq(Sys,Exp,Opt);
end

At the end, we plot the results. The ENDOR frequencies are scaled with the Larmor frequency of the nucleus.

NZ = larmorfrq('14N',Exp.Field);
subplot(1,2,1); plot(eeQq/NZ,sort(p/NZ).','k');
axis tight, xlabel('eeQq/wnuc'); ylabel('frequency/wnuc');
subplot(1,2,2); plot(eeQq/NZ,sort(w).','k');
axis tight, ylabel('amplitude'); xlabel('eeQq/wnuc');