EasySpin can calculate magnetic moments and magnetic susceptibilities for arbitrary spin systems. Experimentally, these are typically obtained using a SQUID magnetometer. This user guide explains how to calculate these quantities.

The following topics are covered:

• Running the calculation
• The spin system
• Experimental parameters
• Specifying the type of output
• Units and different representations of the susceptibility
• Single crystals
• Calculation options
• Least-squares fitting

To calculate magnetic moments and magnetic susceptibilities, you need to provide information on the sample (spin system) and the fields and temperatures used in the measurement. With these, you then call the function curry. By default, it returns one or two outputs, but this can be customized:

muBM = curry(Sys,Exp);
[muBM,chimol] = curry(Sys,Exp);

Sys contains the spin system, and Exp contains the experimental parameters. muBM is the z component of the magnetic dipole moment in units of Bohr magnetons, and chimol is the zz component of the molar magnetic susceptibility. z indicates the direction along the external magnetic field.

Additionally, you can provide a third input argument, Opt, that is a structure containing calculation options such as which output to provide and with system of units (SI or CGS) to use. For example,

Opt.Units = 'CGS';           % use CGS-emu units
Opt.Output = 'chimol';       % calculate molar magnetic susceptibility
chimol = curry(Sys,Exp,Opt);

To calculate magnetic moment and magnetic susceptibility, first of all you need to provide the system of spins and the parameters of the associated spin Hamiltonian as a spin system structure. Here is a simple example of two coupled spins-1/2 with isotropic g factors and strong exchange coupling:

Sys.S = [1/2 1/2];
Sys.g = [2 2];
Sys.J = -2*-10*29.98e3;   % MHz

Be mindful that EasySpin's isotropic electron-electron coupling term Sys.J uses the convention +J S₁S₂. Other conventions are often used in the literature. See here for details.

Also, EasySpin expects all couplings in units of MHz. Very often, exchange couplings are given in cm^-1. To convert cm^-1 to MHz use

J_cm = 5;              % value in cm^-1
J_MHz = J_cm*29.98e3;  % cm^-1 -> MHz conversion
Sys.J = J_MHz;         % for H = +J*S1*S2

Sys.J = 5 * 29.98e3;   % all in one line

You can also give antisymmetric and symmetric coupling terms, in Sys.dvec and Sys.eeD, for details see the page on spin system structures.

Often a term called temperature-independent paramagnetism (TIP or χ_TIP) is included to model the magnetic susceptibility. It is simply added to χ_zz.

TIP_CGSemu = 5e-4;              % value in cm^3 mol^-1
TIP_SI = TIP_CGSemu*4*pi*1e-6;  % conversion from CGS-emu to SI units
Sys.TIP = TIP_SI;               % EasySpin expect TIP to be in SI units

Sys.TIP = 5e-4*4*pi*1e-6;   % all in one line

As with the rest of EasySpin, there is in principle no limit on the number of spins you can specify or on how they can be coupled. The only limit is the computer memory required to hold the spin Hamiltonian and operator matrices.

For SQUID magnetometry, you only need two basis experimental settings: the magnetic field values and the temperatures at which you measure the magnetic moment and the magnetic susceptibility.

Give the temperature in Exp.Temperature, in units of K. You can provide either a single temperature or an array of temperatures.

Exp.Temperature = 298;    % room temperature
Exp.Temperature = 4:300;  % large temperature range

For the magnetic field, use Exp.Field and provide the field values in units of mT. Again, you can give either a single value, or an array of values. If the experiment is run at zero magnetic field, you can omit Exp.Field.

Exp.Field = 3000;       % 3000 mT = 3 T
Exp.Field = 0:10:6000;  % 0 to 6T range

If you give a single value for both the temperature and the field, curry calculates single values for the outputs. If you provide a single value for one of the parameters and an array for the other, you get an array of results. For example, providing Exp.Field = 1000; and Exp.Temperature = 4:300 will yield curves as a function of temperature. Finally, if you provide arrays for both field and temperature, curry returns 2D arrays for the results.

Most SQUID measurement protocols do not use the same field and temperature steps for measurements of the magnetic moment and magnetic susceptibility. To simulate this, call curry twice:

% calculate magnetic moment for a few temperatures over full field range
Exp.Temperature = [2, 5, 7,10];  % few temperatures (K)
Exp.Field = 0:10:6000;         % full field range (mT)
Opt.Output = 'muBM';
muBM = curry(Sys,Exp,Opt)

% calculate magnetic susceptibility for a few fields over full temperature range
Exp.Temperature = 4:300;    % large temperature range (K)
Exp.Field = [0, 500];       % two field values (mT)
Opt.Output = 'chimol';
chimol = curry(Sys,Exp,Opt)

Very often, magnetic susceptibility is understood to be in zero field. However, to measure the derivative of the magnetization with respect to the magnetic field (that is what the magnetic susceptibility is) a magnetic field is needed. For various experimental reasons this field is often not infinitesimally small but can be quite substantial. Therefore, if larger fields are used it might be more correct to use the following:

m4chi =  curry(Sys,Exp,Opt);          % magnetic moment at the field used in the experiment
% divide magnetic moment by magnetic field and unit conversion
chisim_cgs = m4chi*avogadro*bmagn*1e2./Exp.Field(:);
chisim_si = (4*pi*1e-6) *chisim_cgs ; % convert to SI units

curry can calculate a variety of quantities related to magnetic susceptibility and magnetic moment. Use Opt.Output to specify what curry should calculate. One or more keywords are possible. A few examples are

Opt.Output = 'muBM';           % single-center magnetic moment in units of Bohr magnetons
Opt.Output = 'chimol';         % molar magnetic susceptibility
Opt.Output = 'chimolT';        % molar magnetic susceptibility times temperature
Opt.Output = 'muBM chimolT';   % calculate both quantities

The number of output arguments of curry must match the number of keywords in Output. The quantity of the first keyword will be in the first output and so on. Be careful:

Opt.Output = 'chimol muBM';         % reverse order to the default
[chizz1,muz1] = curry(Sys,Exp,Opt); % Opt.Output is given, output in reverse order to the default
[muz2,chizz2] = curry(Sys,Exp);     % Opt.Output is omitted, Output in default order

Several ways to represent the magnetic susceptibility are used. Beside χ_zz (where often the index is omitted), the product of temperature and susceptibility χ_zzT and the inverse of the susceptibility 1/χ_zz are common. Also the effective magnetic moment μ_eff is used. All of them contain the same information, however their appearance is quite different. A free paramagnet is described by χ_zz = C/T with Curie constant C and temperature T. Hence, χ_zz approaches C at low temperatures and tend towards zero at high temperatures. In contrast 1/χ_zz decrease linearly with decreasing temperature. χ_zzT is simply a constant line at C. In the section Least-squares fitting we will discuss implications of that.

Note that curry does not calculate volume or mass magnetic susceptibilities.

In magnetometry, there are two systems of units in use, the modern SI system and the old CGS-emu system. curry can return results in either of the two. By default, the SI system is used. Specify the system of units directly in the options field Opt.Units.

Opt.Units = 'SI';     % use SI units
out = curry(Sys,Exp,Opt);

Opt.Units = 'CGS';    % use CGS-emu units
out = curry(Sys,Exp,Opt);

The SI unit for the magnetic moment is J T^-1, whereas its CGS-emu unit is erg G^-1.

curry returns the calculated magnetic moments in what might be called "natural units". The numerical value returned by curry can be taken either as the molecular magnetic moment μ_z (magnetic moment per molecule) in units of Bohr magnetons (μ_B), or as the molar magnetic moment μ_mol,z (magnetic moment per mole) in "molar Bohr magnetons" (N_Aμ_B). These two are numerically identical. For example, muz = 0.53 means the molecular magnetic moment μ_z is 0.53 μ_B and the molar magnetic moment μ_mol,z is 0.53 N_Aμ_B.

The SI units for these two magnetic moment quantities are J T^-1 and J T^-1 mol^-1. To convert from the natural units to the SI units, use EasySpin's functions for the Avogadro number (avogadro) and the Bohr magneton (bmagn):

muz_SI = muzBM*bmagn;          % molecular magnetic moment, natural units -> SI units
muzmol_SI = muz_SI*avogadro    % molecular magnetic moment -> molar magnetic moment

The single-particle (single-molecule, atomic) magnetic susceptibility has the SI unit m³ mol^-1 and the CGS-emu unit cm³ mol^-1. The molar magnetic susceptibility has the SI unit m³ mol^-1 and the CGS-emu unit cm³ mol^-1. To convert these susceptibilities between SI and CGS-emu, use

chi_SI = chi_CGS*(4*pi*1e-6);        % conversion from CGS-emu to SI units
chimol_SI = chimol_CGS*(4*pi*1e-6);  % conversion from CGS-emu to SI units

chi_CGS = chi_SI/(4*pi*1e-6);        % conversion from SI to CGS-emu units
chimol_CGS = chimol_SI/(4*pi*1e-6);  % conversion from SI to CGS-emu units

curry can also calculate magnetic moment and molar magnetic susceptibility of single crystals. For this, provide three things: (a) the orientation of the crystal in the spectrometer in Exp.SampleFrame, (b) the space group of the crystal in Exp.CrystalSymmetry, and (c) the orientation of the molecular frame relative to the crystal frame in Exp.MolFrame. You can omit the latter two, in which case the space group is assumed to be P1, and the molecular frame is assumed to be aligned with the crystal frame. You can't omit Exp.SampleFrame.

For more information on the various frames and how to specify the relative orientations, see the description of frames.

Here are some examples on how to use this.

Exp.CrystalSymmetry = 'P212121';     % space group

Exp.SampleFrame = [0 0 0];           % crystal z axis along magnetic field
Exp.SampleFrame = [0 -pi/2 0];       % crystal x axis along magnetic field
Exp.SampleFrame = [-pi/2 -pi/2 0];   % crystal y axis along magnetic field

Without information about sample orientation (Exp.SampleFrame), curry assumes the sample is a powder and performs a powder average. The powder average involves evaluating many orientations of the paramagnetic center with respect to the magnetic field, and then averaging them. To adjust the number of orientations included in the powder average, use Opt.GridSize. The higher the number, the more orientations are included.

Opt.GridSize = 10;   % 10° increments
Opt.GridSize = 19;   % 5° increments
Opt.GridSize = 31;   % 3° increments
Opt.GridSize = 91;   % 1° increments

A sufficient number of orientations is necessary to obtain correct data. If you are not sure whether you have enough orientations, increase Opt.GridSize until the calculated data do not change any more.

Of course, if the spin Hamiltonian is isotropic, you do not have to run a powder average. curry detects this situation automatically.

You can use EasySpin's least-squares fitting tool esfit to fit magnetometry data using curry.

In almost all cases magnetometry data is given in absolute units, therefore it is definitely NOT recommended to scale data before least-square fitting. Use FitOpt.AutoScale = false;.

For most spin systems it is not recommended to fit χ_zz. Instead χ_zzT (or μ_eff) should be used, especially when g factors should be obtained. This can be easily visualized by converting poor fits to χ_zzT (or μ_eff) to χ_zz and compare it to the data or you run the following example:

clear, clc

Sys1.S = 2;
Sys1.g = [2.1 1.9];  % axial g
Sys1.D = 1*clight*1e-4;  % cm^-1 -> MHz
Sys2 = Sys1;
Sys2.g = 1.9;  % isotropic g

Exp.Field = 10;  % mT
Exp.Temperature = 1:300;  % K
Opt.Output = 'chimolT chimol';
Opt.Units = 'CGS';
[chiT1,chi1] = curry(Sys1,Exp,Opt);
[chiT2,chi2] = curry(Sys2,Exp,Opt);

subplot(2,1,1)
plot(Exp.Temperature,[chiT1;chiT2])
title('\chi T');
subplot(2,1,2)
plot(Exp.Temperature,[chi1;chi2])
title('\chi');

To increase the reliability of the least-square fitting it is often a good idea to fit both the magnetic moment and the magnetic susceptibility simultaneously. Let's say you have measured χ_zzT at at two fields and a range of temperatures, and the magnetic moment at 3 temperatures over a range of fields:

clear, clc
Sys.S = 2;
Sys.g = 2;
Sys.D = 10*30e3;  % cm^-1 -> MHz

Exp.Temperature = [2:0.5:20 25:5:300];  % K
Exp.Field = [100,500];  % mT
Opt.Output = 'chimolT';
chimolTdata = curry(Sys,Exp);

subplot(2,1,1)
plot(Exp.Temperature,chimolTdata);

Exp.Temperature = [2,5,10];  % K
Exp.Field = 0:100:7000;  % mT
Opt.Output = 'muBM';
muBMdata = curry(Sys,Exp,Opt);

subplot(2,1,2)
plot(Exp.Field,muBMdata);

chimolTdata should be a 2 x 93 matrix and muBMdata a 71 x 3 matrix.

You can fit both measured quantities simultaneously by defining your own custom model function that calls curry twice:

Sys.D = 7*30e3;
vSys.D = 5*30e3;

% No scaling! Assuming correct diamagnetic corrections and sample weight measure,
% the absolute value contains valuable information!
FitOpt.AutoScale = false;

% Least-squares fitting
data = [5*chimolTdata(:); muBMdata(:)];
esfit(data,@myfun,{Sys,Exp,Opt},{vSys},FitOpt);

function datasim = myfun(Sys,Exp,Opt)
Exp.Temperature = [2:0.5:20 25:5:300];  % K
Exp.Field = [100,500];  % mT
Opt.Output = 'chimolT';
chimolTsim = curry(Sys,Exp);

Exp.Temperature = [2,5,10];  % K
Exp.Field = 0:100:7000;  % mT
Opt.Output = 'muBM';
muBMsim = curry(Sys,Exp,Opt);

datasim = [5*chimolTsim(:); muBMsim(:)];
end

For more details on least-squares fitting, see the corresponding user guide and the reference page on esfit. Also, there is an example for fitting magnetometry data.