\(\renewcommand\AA{\unicode{x212B}}\)

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

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

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).

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']))

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 (non-fitting parameters)

Name	Type	Default	Description
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 (fitting parameters)

Name	Default	Description
BmolX	0.0	The x-component of the molecular field.
BmolY	0.0	The y-component of the molecular field.
BmolZ	0.0	The z-component of the molecular field.
BextX	0.0	The x-component of the external field.
BextY	0.0	The y-component of the external field.
BextZ	0.0	The z-component of the external field.
B20	0.0	Real part of the B20 field parameter.
B21	0.0	Real part of the B21 field parameter.
B22	0.0	Real part of the B22 field parameter.
B40	0.0	Real part of the B40 field parameter.
B41	0.0	Real part of the B41 field parameter.
B42	0.0	Real part of the B42 field parameter.
B43	0.0	Real part of the B43 field parameter.
B44	0.0	Real part of the B44 field parameter.
B60	0.0	Real part of the B60 field parameter.
B61	0.0	Real part of the B61 field parameter.
B62	0.0	Real part of the B62 field parameter.
B63	0.0	Real part of the B63 field parameter.
B64	0.0	Real part of the B64 field parameter.
B65	0.0	Real part of the B65 field parameter.
B66	0.0	Real part of the B66 field parameter.
IB21	0.0	Imaginary part of the B21 field parameter.
IB22	0.0	Imaginary part of the B22 field parameter.
IB41	0.0	Imaginary part of the B41 field parameter.
IB42	0.0	Imaginary part of the B42 field parameter.
IB43	0.0	Imaginary part of the B43 field parameter.
IB44	0.0	Imaginary part of the B44 field parameter.
IB61	0.0	Imaginary part of the B61 field parameter.
IB62	0.0	Imaginary part of the B62 field parameter.
IB63	0.0	Imaginary part of the B63 field parameter.
IB64	0.0	Imaginary part of the B64 field parameter.
IB65	0.0	Imaginary part of the B65 field parameter.
IB66	0.0	Imaginary part of the B66 field parameter.

Categories: FitFunctions | General

Source

C++ header: CrystalFieldHeatCapacity.h

C++ source: CrystalFieldHeatCapacity.cpp