CrystalFieldMagnetisation.rst

CrystalFieldMagnetisation

.. index:: CrystalFieldMagnetisation

Description

This function calculates the crystal field (molar) magnetic moment as a function of applied magnetic field in a specified direction, in either atomic (\mu_B//ion), SI (Am²/mol) or cgs (erg/Gauss/mol == emu/mol) units. If using cgs units, the magnetic field (x-axis) is expected to be in Gauss. If using SI or atomic units, the field should be given in Tesla.

Strictly, to obtain the magnetisation, one should divide by the molar volume of the material.

Theory

The function calculates the expectation value of the magnetic moment operator \mathbf{\mu} = g_J \mu_B \mathbf{J}:

M(B) = \frac{1}{Z} \sum_n \langle V_n(H) | g_J \mu_B \mathbf{J} | V_n(H) \rangle \exp(-\beta E_n(H))

where B is the magnetic field in Tesla, g_J is the Landé g-factor, \mu_B is the Bohr magneton. 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. V_n(B) and E_n(B) are the n^th eigenvector and eigenvalue (wavefunction and energy) obtained by diagonalising the Hamiltonian:

\mathcal{H} = \mathcal{H}_{\mathrm{cf}} + \mathcal{H}_{\mathrm{Zeeman}} = \sum_{k,q} B_k^q \hat{O}_k^q
- g_J \mu_B \mathbf{J}\cdot\mathbf{B}

where in this case the magnetic field \mathbf{B} is not normalised. Finally, \beta = 1/(k_B T) with k_B the Boltzmann constant and T the temperature, and Z is the partition sum Z = \sum_n \exp(-\beta E_n(H)).

Example

Here is an example of how to fit crystal field parameters to a magnetisation measurement. All parameters disallowed by symmetry are fixed automatically. The "data" here is generated from the function itself.

The x-axis is given in Tesla, and the magnetisation (y-axis) is in bohr magnetons per magnetic ion (\mu_B/ion).

.. testcode:: ExampleCrystalFieldMagnetisation

    import numpy as np

    # Build a reference data set
    fun = 'name=CrystalFieldMagnetisation,Ion=Ce,B20=0.37737,B22=0.039770,B40=-0.031787,B42=-0.11611,B44=-0.12544,'
    fun += 'Temperature=10'

    # This creates a (empty) workspace to use with EvaluateFunction
    x = np.linspace(0, 30, 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=CrystalFieldMagnetisation,Ion=Ce,Symmetry=C2v,Temperature=10,B20=0.4,B22=0.04,B40=-0.03,B42=-0.1,B44=-0.1,'
    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:: ExampleCrystalFieldMagnetisation

.. testoutput:: ExampleCrystalFieldMagnetisation
   :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 =  0.39541
    B22 =  0.030001
    B40 = -0.029841
    B42 = -0.11611
    B44 = -0.1481
Cost function value =  1.2987e-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
   Temperature;Double;1.0; Temperature in Kelvin of the measurement.
   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/ion), 'SI' (Am^2/mol) or 'cgs' (erg/G/mol).

.. properties::

.. categories::

.. sourcelink::