CrystalFieldMoment¶

Description¶

This function calculates the crystal field contribution to the magnetic moment as a function of temperature at a constant applied magnetic field in a specified direction, in either atomic ($\mu_B/$/ion), SI (Am²/mol) or cgs (erg/Gauss/mol == emu/mol) units.

Theory¶

The magnetic moment is calculated as the thermal 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, and $\mu_B$ 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 M(T) to a measured dataset. All parameters disallowed by symmetry are fixed automatically. The “data” here is generated from the function itself.

The x-axis is the temperature in Kelvin, and the magnetic moment (y-axis) is in Am²/mol (SI units), and the “measurement” was done with a field of 0.01 Tesla along the [110] direction of the crystal field (not necessarily the crystallographic [110] direction).

import numpy as np

# Build a reference data set
fun = 'name=CrystalFieldMoment,Ion=Ce,B20=0.37737,B22=0.039770,B40=-0.031787,B42=-0.11611,B44=-0.12544,'
fun += 'Hmag=0.01, Hdir=(1,1,0), Unit=SI,'

# 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=CrystalFieldMoment,Ion=Ce,Symmetry=C2v,B20=0.37,B22=0.04,B40=-0.032,B42=-0.12,B44=-0.13,'
fun += 'Hmag=0.01, Hdir=(1,1,0), Unit=SI,'
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 =  0.37745
    B22 =  0.016732
    B40 = -0.032093
    B42 = -0.11298
    B44 = -0.12685
Cost function value =  9.7067e-18

Attributes (non-fitting parameters)¶

Properties (fitting parameters)¶

Categories: FitFunctions | General

Source¶

C++ header: CrystalFieldMoment.h

C++ source: CrystalFieldMoment.cpp