Defining a spin system

Gives the electron spin quantum number(s). An arbitrary number of electron spins can be specified. Examples:

Sys.S = 1/2;            % one electron spin with S=1/2
Sys.S = 5/2;            % an S=5/2 spin
Sys.S = [1/2, 1/2];     % two S=1/2 spins
Sys.S = [1, 1, 1/2];    % two S=1 spins and one S=1/2 spin

If S is not given, it is automatically set to 1/2.

A string containing a comma-separated list of nuclear isotopes specifying the nuclear spins in the spin system. An arbitrary number of nuclei can be specified. See also the functions nucgval and nucspin.

Sys.Nucs = '1H';              % one hydrogen
Sys.Nucs = '63Cu';            % a 63Cu nucleus
Sys.Nucs = '59Co,14N,14N';    % a 59Co and two 14N nuclei

If there are no nuclear spins in the system, don't specify this field or set it to '' (an empty string).

It is also possibe to specify elements instead of specific isotopes.

Sys.Nucs = 'Cu';                % a natural-abundance mixture of 63Cu and 65Cu
Sys.Nucs = 'Cu,14N';            % Cu plus a pure 14N

If not a single isotope, but a natural-abundance mixture of isotopes is needed, just omit the mass number. EasySpin will automatically generate all possible isotopologues (isotopes combinations) and simulate all associated spectra. You can freely mix single isotopes and natural-abundance mixtures in one spin system.

Sys.Nucs = 'Cu';          % a mixture of 69% 63Cu and 31% 65Cu
Sys.Nucs = 'Cu,14N';      % same as above, plus a 14N
Sys.Nucs = 'F,C';         % 100% 19F, plus a mixture of 99% 12C and 1% 13C

Depending on whether the g tensors are orthorhombic, axial or isotropic, different ways of input are possible:

• Orthorhombic: Each row contains the three principal values of the g tensor of one electron spin.

  Sys.g = [2 2.05 2.3];                % orthorhombic g, for one electron spin
  Sys.g = [2 2.1 2.3; 1.9 1.95 2.01];  % orthorhombic g, for two electron spins
  

• Axial: For axial g tensors, only two values are needed. The first value is the one perpendicular to the easy axis, and the second is the one parallel to it.

  Sys.g = [2.25 2.03];                % axial g, for one electron spin
                                      % = [2.25 2.25 2.03]
  Sys.g = [2.25 2.03; 1.99 1.98];     % axial g, for two electron spins
                                      % = [2.25 2.25 2.03; 1.99 1.99 1.98]
  

• Isotropic: If g is isotropic, it is sufficient to give its isotropic value once.

  Sys.g = 2.005;                      % isotropic g, for one electron spin
  Sys.g = [2.0023; 2.0025];           % isotropic g, for two electron spins
  

• No value: If no values are given, EasySpin assumes isotropic g values of 2.0023... (see gfree) for all electron spins.

Each row of this array contains the three Euler angles (in radians) for the passive rotation which transforms the g matrix of the associated electron spin from its eigenframe to the molecular frame (see also the function erot). If not specified, gpa is assumed to be all-zero, that is, all tensors are aligned with the molecular frame.

Sys.gpa = [0 10 0]*pi/180;                % one electron spin
Sys.gpa = [0 0 0; 0 pi/4 0; 0 -pi/4 0];   % three electron spins

With these angles, EasySpin can transform a g tensor from its diagonal eigenframe representation to the molecular frame representation. Here is an explicit example how this is done internally:

gpa = [10 34 -2]*pi/180;  % Euler angles, in radians
g = [2.1 1.97 2.04];      % principal values

R = erot(gpa);        % rotation matrix
gdiag = diag(g)       % g in eigenframe
gM = R*gdiag*R.'      % g in molecular frame

which gives the following output

gdiag =
    2.1000         0         0
         0    1.9700         0
         0         0    2.0400
gM =
    2.0797   -0.0147    0.0264
   -0.0147    1.9728   -0.0115
    0.0264   -0.0115    2.0575

Principal values of the hyperfine interaction matrices A/h, in MHz. Each row contains the principal values of the A tensors of one nucleus. A(k,:) refers to nuclear spin k. Orientations of the A matrices are given in Apa.

Sys.A = [-6 12 23];            % 1 electron and 1 nucleus
Sys.A = [10 10 -20; 30 40 50]; % 1 electron and 2 nuclei

For axial and isotropic hyperfine tensors, the notation can be shortened, just as in the case of the g tensor.

Sys.A = [4 10];       % = [4 4 10]  (axial, 1 electron and 1 nucleus)
Sys.A = 34;           % = [34 34 34] (isotropic, 1 electron and 1 nucleus)
Sys.A = [4 10; 1 2];  % = [4 4 10; 1 1 2]  (axial, 1 electron and 2 nucleui)
Sys.A = [7; 3];        % = [7 7 7; 3 3 3] (isotropic, 1 electron and 2 nuclei)

If the system contains more than one electron spin, each row contains the principal values of the hyperfine couplings to all electron spins, listed one after the other.

Sys.A = [10 10 -20 30 40 50];                % 2 electrons and 1 nucleus
Sys.A = [10 10 -20 30 40 50; 1 1 -2 3 4 5];  % 2 electrons and 2 nuclei

It is possible to specify full A matrices in A. The 3x3 matrices have to be combined like the 1x3 vectors used when only principal values are defined: For different electrons, put the 3x3 matrices side by side, for different nuclei, on top of each other. If full matrices are given in A, Apa is ignored.

Sys.A = [5 0 0; 0 5 0; 0 0 5]                          % 1 electron and 1 nucleus
Sys.A = [[5 0 0; 0 5 0; 0 0 5]; [10 0 0; 0 10 0; 0 0 10]]     % 1 electron and 2 nuclei
Sys.A = [[5 0 0; 0 5 0; 0 0 5], [10 0 0; 0 10 0; 0 0 10]]     % 2 electrons and 1 nucleus

Spherical form of the hyperfine matrix, in terms of its isotropic, axial and rhombic components. The units are MHz. A and A_ cannot be used at the same time.

Sys.A_ = 2;         % isotropic component only
Sys.A_ = [2 3]      % isotropic and axial component
Sys.A_ = [2 3 0.4]  % isotropic, axial and rhombic component

A = A_(1) + [-1,-1,2]*A_(2) + [-1,+1,0]*A_(3);

Array of Euler angles giving the orientations of the various A matrices in the molecular frame, as described above for gpa. If Apa is not specified, it is assumed to be all-zero, that is, all tensors are aligned with the molecular frame. See also erot. Apa has a layout analogous to A. Each row contains the three Euler angles for one nucleus.

Sys.Apa = [0 20 0]*pi/180;  % one electron spin, one nucleus
Sys.Apa = [0 0 0; 0 10 90; 12 -30 34]*pi/180 % one electron spin, three nuclei

Sys.Apa = [0 20 0, 13 -30 80]*pi/180;               % 2 electrons, 1 nucleus
Sys.Apa = [0 20 0, 0 0 0; 0 10 0, 0 30 50]*pi/180;  % 2 electrons, 2 nuclei

Array specifying the principal values of the quadrupole tensors Q/h for all nuclei, in MHz. There are several possible input ways.

• For axial Q tensor: Specify eeQq/h for each nucleus

  Sys.Q = 0.7            % one nucleus, eeQq/h = 0.7 MHz
  Sys.Q = [0.7, 1.2]     % same for two nuclei
  

• For orthorhombic Q tensor: Specify eeQq/h and eta for each nucleus, two numbers per row.

  Sys.Q = [1.2 0.29]         % one nucleus, eeQq/h = 1.2 MHz and eta = 0.29
  Sys.Q = [1.2 0.29; 0.1 0]  % same with a second nucleus
  

• For all Q tensors: specify three principal values of the Q tensor, for one nucleus per row.

  Sys.Q = [-1 -1 2]                % one nucleus, Qx = -1, Qy = -1, Qz = +2 MHz
  Sys.Q = [-1 -1 2; 0.3 0.2 -0.5]  % same with a second nucleus
  

eeQq = 1; eta = 0.2;
I = 1;         % nuclear spin must be known!
Q = eeQq/(4*I*(2*I-1)) * [-1+eta, -1-eta, 2]

See also the reference page on the nuclear quadrupole interaction.

Array of Euler angles describing the orientation of the Q tensors in the molecular frame, analogous to gpa and Apa. There must be one row of three angles for each nucleus.

Sys.Qpa = [0 pi/4 0];                    % one nucleus
Sys.Qpa = pi/180*[0 30 0; 0 -30 0];      % two nuclei

n-vector or nx2 array or nx3 array of real

Values, in MHz, that specify the zero-field splitting of the electron spins, one row per electron. Each row of D can contain two, one, or three values. If two values are given, they are taken to be the D and the E parameter. If only one value is given, E is assumed to be zero. If three values are given, they are taken to represent the three principal values of the D tensor.

% Examples for one electron spin
Sys.S = 5/2;
Sys.D = [120 15];       % D = 120 MHz, E = 15 MHz
Sys.D = 120;            % D = 120 MHz, E = 0 MHz
Sys.D = [-25 -55 80];   % principal values of the D tensor, MHz

If a system with more than one electron spin is given, the zero-field splitting for each electron has to be given. For electron spins with S=1/2, zeros should be given.

% Examples for three electron spins
Sys.S = [1,1,1/2]; 
Sys.D = [120 100 0];                      % D parameters only
Sys.D = [120 15; 90 10; 0 0];             % D and E parameters
Sys.D = [-25 -55 80; -20 -40 60; 0 0 0];  % principal values of the D tensors

In terms of D and E, the principal values of the D tensor are given by

D = 120; E = 15;                     % D and E parameters
Dpv = [-1,-1,2]/3*D + [1,-1,0]*E     % conversion to D principal values

nx3 matrix of real

Euler angles describing the orientation of the D tensors, completely analogous to gpa and Qpa. If absent, it is assumed to be all zeros.

Internally, EasySpin uses the following procedure to compute the full D tensor in the molecular frame from the given principal values and the Euler angles

Dpv = [-25 -55 80];      % sample principal values
Dpa = [10 20 0]*pi/180;  % sample tilt angles

R = erot(Dpa)            % rotation matrix
D = R*diag(Dpv)*R.'      % computation of the full D tensor

1x2 array of real

Values, in MHz, of the fourth-order parameters a and F. For their definition, see the reference page on high-order operators.

Sys.aF = [10 -3];   % in MHz

Both a and F are defined in terms the molecular reference frame, which is assumed to coincide the four-fold symmetry axes of the cubic part. For other orientations (e.g. along the trigonal axes), the high-order parameters B40 and B44 can be used.

real or 1x2 array

The coefficients for the extended Stevens operators of the electron spin (see also the function stev). B20, B40 and B60 are scalars. The other fields specify two parameters each. E.g. B43 represents . If only one element is given, it is taken as .

1xN or Nx3 or 3Nx3 array of real

Principal value of the electron-electron interaction matrices. Each row corresponds to the diagonal of an interaction matrix (in its eigenframe). They are lexicographically ordered according to the indices of the electrons involved. If n is the number of electrons, there are N = n(n-1)/2 rows. E.g. for four electrons there have to be six rows with the principal values for the interaction of electrons 1-2, 1-3, 1-4, 2-3, 2-4, and 3-4, in this order. For two electrons, ee contains one row only.

E.g. for two S=1/2, the electron-electron interaction with isotropic and dipolar coupling is Sys.ee = Jiso + Jdip*[1 1 -2].

Sys.S = [1/2 1/2];     % two electrons
Sys.ee = [50 50 100];  % one coupling matrix

Sys.S = [1/2 1/2 1/2];                   % three electrons
Sys.ee = [50 50 100; 10 10 -30; 0 0 0];  % three coupling matrices

If only isotropic coupling is wanted, one number per coupling is sufficient.

Sys.S = [1/2 1/2];   % two electrons
Sys.ee = 1000;       % one coupling

Sys.S = [1/2 1/2 1/2 1/2];     % four electrons
Sys.ee = [10 0 0 5 -10 20];    % six couplings (1-2,1-3,1-4,2-3,2-4,3-4)

For accomodating antisymmetric exchange, the full 3x3 interaction matrices (instead of the 3 principal values and 3 Euler angles) can be specified. For a 2-electron system, ee is then a 3x3 array. For more electrons, the 3x3 matrices are stacked on top of each other.

Sys.S = [1/2 1/2];                   % two electrons
Sys.ee = [50 0 0;0 50 0; 0 0 100];   % one coupling matrix

Sys.S = [1/2 1/2 1/2];                   % three electrons
ee12 = [50 0 0; 0 50 0; 0 0 100];
ee13 = diag([10 10 -30]);
ee23 = zeros(3);
Sys.ee = [ee12;ee13;ee23];               % three coupling matrices

Nx3 array of real

Euler angles describing the orientation of the electron-electron interaction matrices specified in ee. Each row contains the Euler angles for the corresponding row in ee.