CrystalFieldSusceptibility.rst

CrystalFieldSusceptibility

.. index:: CrystalFieldSusceptibility

Description

This function calculates the crystal field contribution to the molar magnetic susceptibility using the Van Vleck formula. The function outputs the results by default in cgs units of cm³/mol == "emu/mol". There are also options to output the result in SI (m³/mol) or "atomic" units (\mu_B/Tesla/ion).

Theory

The magnetic susceptibility can be calculated by treating the magnetic field (Zeeman interaction) as a perturbation on the crystal field energy. To second order, the susceptibility per mole of magnetic ion is given by:

\chi(T) = \frac{N_A}{Z} \sum_n \left[ \frac{| \langle V_n | g_J \mu_B \mathbf{J} | V_n \rangle | ^2}{k_B T}
- 2 \sum_{m \neq n} \frac{| \langle V_n | g_J \mu_B \mathbf{J} | V_m \rangle | ^2}{E_n - E_m} \right] \exp(-\beta E_n)

where N_A is Avogadro's constant, k_B is Boltzmann's constant, Z is the partition sum, and V_n and E_n are the n-th wavefunction (eigenvector) and energy level (eigenvalue) of the unperturbed crystal field Hamiltonian. g_J is the Landé g-factor, \mu_B is the Bohr magneton and the moment operator is defined as \mathbf{J} = \hat{J}_x B_x + \hat{J}_y B_y + \hat{J}_z B_z where \hat{J}_x, \hat{J}_y, and \hat{J}_z are the angular momentum operators in Cartesian coordinates, with z defined to be along the quantisation axis of the crystal field (which is usually defined to be the highest symmetry rotation axis). B_x, B_y, and B_z are the components of the unit vector pointing in the direction of the applied magnetic field in this coordinate system.

Finally, in order to account for the effect of any exchange interactions in the system which will shift the susceptiblity curve up or down (analogous to the Curie-Weiss temperature), and any residual (background) susceptibility in the sample (perhaps from an impurity), the actual magnetic susceptibility calculated by this function is:

\chi^{\mathrm{eff}} = \frac{\chi(T)}{1 - \lambda \chi(T)} + \chi_0

where \lambda parameterises an effective exchange interaction with \chi the bare (paramagnetic Crystal Field) susceptibility, and \chi_0 the residual susceptibility. A negative \lambda indicates overall antiferromagnetic interactions, whilst a positive \lambda corresponds to overall ferromagnetic interactions.

Example

Here is an example of how to the crystal field parameters to a susceptibility dataset. All parameters disallowed by symmetry are fixed automatically. The "data" here is generated from the function itself, for a field along the [111] direction with respects to the crystal field parameters (not necessarily the [111] crystallographic direction).

The x-axis is given in Kelvin, and the susceptibility (y-axis) is in cgs units of cm³/mol (==emu/mol).

.. testcode:: ExampleCrystalFieldSusceptibility

    import numpy as np

    # Build a reference data set
    fun = 'name=CrystalFieldSusceptibility,Ion=Ce,B20=0.37737,B22=0.039770,B40=-0.031787,B42=-0.11611,B44=-0.12544,'
    fun += 'Hdir=(1,1,1), Unit=cgs, inverse=1,'

    # This creates a (empty) workspace to use with EvaluateFunction
    x = np.linspace(1, 300, 300)
    y = x * 0
    e = y + 1
    ws = CreateWorkspace(x, y, e)

    # The calculated data will be in 'data', WorkspaceIndex=1
    EvaluateFunction(fun, ws, OutputWorkspace='data')

     # Change parameters slightly and fit to the reference data
    fun = 'name=CrystalFieldSusceptibility,Ion=Ce,Symmetry=C2v,B20=0.4,B22=0.04,B40=-0.03,B42=-0.1,B44=-0.1,'
    fun += 'Hdir=(1,1,1), Unit=cgs, inverse=1,'
    fun += 'ties=(B60=0,B62=0,B64=0,B66=0,BmolX=0,BmolY=0,BmolZ=0,BextX=0,BextY=0,BextZ=0)'

    # (set MaxIterations=0 to see the starting point)
    Fit(fun, 'data', WorkspaceIndex=1, Output='fit',MaxIterations=100, CostFunction='Unweighted least squares')
    # Using Unweighted least squares fit because the data has no errors.

    # Extract fitted parameters
    parws = mtd['fit_Parameters']
    for i in range(parws.rowCount()):
        row = parws.row(i)
        if row['Value'] != 0:
            print("%7s = % 7.5g" % (row['Name'], row['Value']))

.. testcleanup:: ExampleCrystalFieldSusceptibility

.. testoutput:: ExampleCrystalFieldSusceptibility
   :hide:
   :options: +ELLIPSIS, +NORMALIZE_WHITESPACE

        B20 =  0...
        B22 =  0...
        B40 = -0...
        B42 = -0...
        B44 = -0...
    Cost function value = ...

Output (the numbers you see on your machine may vary):

    B20 =  0.37737
    B22 =  0.039788
    B40 = -0.031787
    B42 = -0.11611
    B44 = -0.12544
Cost function value =  1.0921e-14

.. attributes::

   Ion;String;Mandatory;An element name for a rare earth ion. Possible values are: Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb.
   Symmetry;String;C1;A symbol for a symmetry group. Setting `Symmetry` automatically zeros and fixes all forbidden parameters. Possible values are: C1, Ci, C2, Cs, C2h, C2v, D2, D2h, C4, S4, C4h, D4, C4v, D2d, D4h, C3, S6, D3, C3v, D3d, C6, C3h, C6h, D6, C6v, D3h, D6h, T, Td, Th, O, Oh
   powder;Boolean;false; Whether to calculate the powder averaged magnetisation or not.
   Hdir;Vector;(0.,0.,1.); The direction of the applied field w.r.t. the crystal field parameters
   Unit;String;'bohr'; The desired units of the output, either: 'bohr' (muB/T/ion), 'SI' (m^3/mol) or 'cgs' (cm^3/mol).
   inverse;Boolean;false; Whether to output 1/chi(T) instead of chi(T).

.. properties::

.. categories::

.. sourcelink::