This tutorial explains how to simulate slow-motion cw EPR spectra of S=1/2 systems using EasySpin's function chili. For full information, see the reference page on chili.
It contains the following topics:chili
Slow-motion cw EPR spectra of S=1/2 systems are computed by the EasySpin function chili.
chili(Sys,Dyn,Exp)
It is called with three arguments. The first argument Sys
tells chili
all about the static parameters of the spin system, and the second argument Dyn
contains parameters of the dynamics of the system.
The third argument Exp
gives the experimental parameters.
If no output argument is given, chili
plots the computed spectrum. But it can also return one or two outputs.
(Don't forget the semicolon at the end of the line to suppress
output to the command window.)
Spec = chili(Sys,Dyn,Exp); [Field,Spec] = chili(Sys,Dyn,Exp);
The outputs Field
and Spec
are
vectors containing the magnetic field axis and the spectrum, respectively.
If these are requested, chili
does not plot the spectrum.
Doing a simulation only requires a few lines of code. A simple example is
Sys = struct('g',[2.008,2.006,2.003],'Nucs','14N','A',[20,20,85]); Dyn = struct('lw',0.3,'tcorr',4e-8); Exp = struct('mwFreq',9.5); chili(Sys,Dyn,Exp);
The first line defines the spin system, a nitroxide radical with anisotropic g and A tensors. The second line gives the dynamic parameters of the radical (a residual linewidth and the rotational correlation time). The third line specifies experimental parameters, here only the microwave frequency (The magnetic field range is chosen automatically). The fourth line calls the simulation function, which plots the result. Copy and paste the code above to your Matlab command window to see the graph.
Of course, the names of the input and output variables don't have
to be Sys
, Dyn
, Exp
, Field
and Spec
.
You can give them any name you like, as long as it is a valid Matlab
variable name, e.g., FremySaltSolution
or QbandExperiment
.
For convenience, thoughout this tutorial, we will use short names like Sys
and Exp
.
The first input argument specifies the static parameters of the paramagnetic molecule. It is a Matlab structure with various fields giving values for the spin system parameters.
Sys.g = [2.008,2.006,2.003]; Sys.Nucs = '14N'; Sys.A = [20,20,80]; % MHz
The first line defined the g tensor of the spin system via its three
principal values. chili
always assumes a single unpaired electron spin S=1/2.
The field Sys.Nucs
contains a string giving all the
magnetic nuclei in the spin system, a nitrogen-14 in the above example.
Use a comma-separated list of isotope labels to give more than one
nucleus. E.g., Sys.Nucs = '14N,1H,1H'
specifies one nitrogen and
two different protons. See the section on multinuclear systems
about details of the slow-motion simulation in that case.
Sys.A
gives the hyperfine couplings in MHz (Megahertz),
with three principal values on a row for each of the nuclei listed in Sys.Nucs
.
The following defines a hydrogen atom with a 10 MHz coupling to the unpaired electron and
a 13C atom with a 12 MHz coupling.
Sys.Nucs = '1H,13C'; Sys.A = [10 12]; % MHz
Remember that chili
(and other EasySpin functions, too),
take the hyperfine coupling values to be in MHz.
Often, values for hyperfine couplings are given in G (Gauss) or mT
(Milltesla), so you have to convert these values.
For g = 2.00232, 1 G corresponds to 2.8025 MHz, and 1 mT corresponds to 28.025 MHz.
The simplest way to convert coupling constants from magnetic field units to MHz is to use the EasySpin
function mt2mhz:
A_MHz = mt2mhz(A_mT); % mT -> MHz conversion A_MHz = mt2mhz(A_G/10); % G -> MHz conversion (1 G = 0.1 mT)
The orientations of the tensors relative to the molecular frame are defined in terms of Euler angles in 3-element array (see the function erot.
Sys.gpa = [0 0 0]; % Euler angles for g tensor Sys.Apa = [0 pi/4 0]; % Euler angles for A tensor
For more details, see the documentation on spin systems.
The second input structure collects parameters relating to the dynamics of the paramagnetic molecules.
There are two possibilies to specify the rate of isotropic rotational diffusion:
either by specifying tcorr
, the rotational correlation time in seconds
Dyn.tcorr = 1e-7; % 100 ns
or by specifying the principal value of the rotational diffusion tensor (in Hertz)
Dyn.Diff = 1e9; % 1 GHzThe two are related via
Diff = 1/6/tcorr;The input field
Diff
can be used to specify an axial rotational diffusion
tensor, by giving a 2-element vector with first the perpendicular and the the parallel
principal value:
Dyn.Diff = [1 2]*1e8; % in Hertz
For the reliability of the simulation algorithm it is strongly recommended to always
use a residual line width to be given in Dyn.lw
Dyn.lw = 10; % MHz
Note that the unit of this residual line width is
chili
is also capable of simulating spectra including Heisenberg spin exchange. The
effective exchange frequency (in Hertz) is specified in Dyn.Exchange
, e.g.
Dyn.Exchange = 1e8; % 100 MHz
A short example of an nitroxide radical EPR spectrum with exchange narrowing is
Nx = struct('g',[2.0088, 2.0061, 2.0027],'Nucs','14N','A',[16 16 86]); Par = struct('mwFreq',9.5,'CenterSweep',[338, 10]); Dyn = struct('tcorr',1e-7,'lw',0.3,'Exchange',1e8); chili(Nx,Dyn,Par);
All experimental settings are given in the second input argument
Exp
, again a Matlab structure.
There are five possible fields:
Exp.mwFreq = 9.5; % GHz, mandatory Exp.CenterSweep = [330 20]; % mT, optional, default: automatic Exp.Range = [320 340]; % mT, optional, default: automatic Exp.Harmonic = 1; % optional, default: 1 Exp.nPoints = 1024; % optional, default: 1024
Exp.mwFreq
gives the spectrometer frequency in GHz. It
is the only field that has to be specified, all others are optional.
There are two ways to specify the magnetic field sweep range.
Either the
center and the sweep width (in mT) are given in Exp.CenterSweep
,
or the lower and upper limit of the sweep range (again in mT) are given in
Exp.Range
.
In many cw EPR spectrometers, the field range is
specified using a center field and a sweep width, so Exp.CenterSweep
is usually the more natural choice.
Exp.CenterSweep
and Exp.Range
are only optional.
If both are omitted, chili
automatically
chooses a field range large enough to accomodate the full spectrum.
If both are given, chili
takes
the values given in Exp.CenterSweep
and ignores those
in Exp.Range
.
The optional field Exp.Harmonic
specifies the detection harmonic. It has three
possible settings:
Exp.Harmonic = 0; % absorption spectrum Exp.Harmonic = 1; % first-derivative spectrum (default) Exp.Harmonic = 2; % second-derivative spectrum
The default value for Exp.Harmonic
is 1. Note that Exp.Harmonic>0
only computes the appropriate derivative of the absorption spectrum. Broadening effects due to strong field modulation have to be added separately, see below.
Yet another optional field is Exp.nPoints
, giving the number of points in the simulated
spectrum. If it is not given, the default 1024 is assumed. You can
set any value, unlike EPR spectrometers, where usually only powers of
2 are available (1024, 2048, 4096, 8192).
The third input structure, Opt
, collects parameters related
to the simulation algorithm.
The field Verbosity
specifies whether chili
should print information about its computation into the Matlab command window.
By default, its value is set to 0, so that chili
is silent.
It can be switched on by giving
Opt.Verbosity = 1; % log information
Another important option is LLKM
. It specifies the number of
orientational basis functions used in the simulation. For spectra in the
fast and intermediate motion regime, the preset values don't have to be
changed. However, close to the rigid limit, the default settings give too
small a number. LLKM
has to be increased:
Opt.LLKM = [24 20 10 10];
To see the values of LLKM
that chili
is using,
set Opt.Verbosity=1
, as described above.
chili
is not capable of simulating a slow-motional cw EPR spectrum of
systems with more than one nucleus.
However, if the hyperfine coupling of one nucleus
is significantly larger than those of the other nuclei, chili
uses an
approximate procedure: The slow-motional spectrum
simulated using only the electron spin and the nucleus with the dominant hyperfine coupling
is convoluted by the isotropic splitting pattern due to all other nuclei. This post-convolution
technique gives resonable results.
A simple example is
CuPc = struct('g',[2.0525 2.0525 2.1994],'Nucs','63Cu,14N','n',[1 4]); CuPc.A = [-54 -54 -608; 52.4 41.2 41.8]; Dynamics = struct('tcorr',10^-7.35,'lw',3); Exp = struct('mwFreq',9.878,'CenterSweep',[330 120],'nPoints',5e3); Opt.LLKM = [36 30 8 8]; chili(CuPc,Dynamics,Exp,Opt);