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 \(\mathbf{\mu} = g_J \mu_B \mathbf{J}\):

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:

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).

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']))

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 (non-fitting parameters)¶

Properties (fitting parameters)¶

Categories: FitFunctions | General

Source¶

C++ header: CrystalFieldMagnetisation.h (last modified: 2020-03-20)

C++ source: CrystalFieldMagnetisation.cpp (last modified: 2020-03-25)