CrystalFieldHeatCapacity.rst

CrystalFieldHeatCapacity

.. index:: CrystalFieldHeatCapacity

Description

This function calculates the magnetic contribution to the heat capacity of a material from the splitting of its electronic energy levels by the crystal field. It is a part of crystal field computation in Mantid and under active development. More documentation will follow as the development progresses.

Theory

The heat capacity at constant volume is given by

C_v = \left. \frac{\partial U}{\partial T} \right|_V = \frac{1}{k_B T^2}
\frac{\partial}{\partial \beta} \left[ \frac{1}{Z}\frac{\partial Z}{\partial \beta} \right]
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \\
= \frac{1}{k_B T^2} \left( \frac{1}{Z}\sum_n E_n^2 \exp(-\beta E_n)
  - \left[ \frac{1}{Z}\sum_n E_n \exp(-\beta E_n) \right]^2 \right)

where k_B is Boltzmann's constant, Z is the partition sum, and E_n is the n-th energy level split by the crystal field. This is obtained by diagonalising the crystal field Hamiltonian.

Example

Here is an example of how to fit function's parameters to a spectrum. All parameters disallowed by symmetry are fixed automatically. The "data" here is generated from the function itself. For real data, you should subtract the phonon contribution manually using either measurements from a phonon blank or a theoretical calculation (e.g. Debye model, or from lattice dynamical calculations) before using it with this function.

The x-axis is given in Kelvin, and the heat capacity (y-axis) is in Joules per mole-Kelvin (Jmol^-1K^-1).

.. testcode:: ExampleCrystalFieldHeatCapacity

    import numpy as np

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

    # 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=CrystalFieldHeatCapacity,Ion=Ce,Symmetry=C2v,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:: ExampleCrystalFieldHeatCapacity

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

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

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

    B20 =  0.40709
    B22 =  0.020272
    B40 = -0.031454
    B42 = -0.10724
    B44 = -0.1314
Cost function value =  4.4642e-15

.. 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

.. properties::

.. categories::

.. sourcelink::