Demagnetizing factors presented by Stoner (1945)

This notebook aims at reproducing the demagnetizing factors values presented by Stoner (1945).

Stoner, E. C.: The demagnetizing factors for ellipsoids, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 36, 803–821, 1945.

Import the required modules



In [1]:

    
from __future__ import division
%matplotlib inline
import numpy as np
import os
from matplotlib import pyplot as plt
from fatiando import utils
import mesher
import prolate_ellipsoid, oblate_ellipsoid, triaxial_ellipsoid



In [2]:

    
# Set some plot parameters
from matplotlib import rcParams
rcParams['figure.dpi'] = 300.
rcParams['font.size'] = 6
rcParams['xtick.labelsize'] = 'medium'
rcParams['ytick.labelsize'] = 'medium'
rcParams['axes.labelsize'] = 'large'
rcParams['legend.fontsize'] = 'medium'
rcParams['savefig.dpi'] = 300.

Stoner (1945, Fig. 4)

Squared eccentricity $\epsilon^{2}$



In [3]:

    
e2 = np.linspace(0.001, 0.99, 100)



In [4]:

    
# m = a/b
m = np.sqrt(1 - e2)

print m









    



[ 0.99949987  0.99448987  0.9894545   0.98439337  0.97930608  0.97419223
  0.96905139  0.96388314  0.95868702  0.95346259  0.94820937  0.94292689
  0.93761464  0.93227212  0.92689882  0.92149418  0.91605765  0.91058867
  0.90508664  0.89955095  0.893981    0.88837611  0.88273565  0.87705891
  0.87134518  0.86559374  0.85980383  0.85397466  0.84810543  0.8421953
  0.8362434   0.83024884  0.82421067  0.81812794  0.81199965  0.80582475
  0.79960217  0.79333079  0.78700943  0.78063688  0.77421188  0.76773312
  0.76119921  0.75460874  0.74796019  0.74125201  0.73448257  0.72765015
  0.72075297  0.71378915  0.70675671  0.69965359  0.69247762  0.6852265
  0.67789782  0.67048904  0.66299748  0.65542029  0.64775447  0.63999684
  0.63214402  0.62419241  0.61613818  0.60797727  0.59970531  0.59131765
  0.58280929  0.57417486  0.56540859  0.55650424  0.54745509  0.53825382
  0.5288925   0.51936247  0.50965427  0.49975752  0.48966078  0.47935141
  0.4688154   0.45803709  0.44699897  0.43568128  0.42406165  0.41211453
  0.39981056  0.38711573  0.37399022  0.360387    0.34624975  0.33151017
  0.31608399  0.29986529  0.28271769  0.26446057  0.24484586  0.22351643
  0.19992423  0.17314675  0.14138564  0.1       ]

Prolate ellipsoids



In [5]:

    
# semi-axes (in m)
a0 = 100
b = a0*m

# demagnetizing factors
n11_prolate = []
n22_prolate = []

for bi in b:
    ellipsoid = mesher.ProlateEllipsoid(0, 0, 0, a0, bi, 29, -50, 180)
    N1, N2 = prolate_ellipsoid.demag_factors(ellipsoid)
    n11_prolate.append(N1)
    n22_prolate.append(N2)

Oblate ellipsoids



In [6]:

    
# semi-axes (in m)
b0 = 100
a = b0*m

# demagnetizing factors
n11_oblate = []
n22_oblate = []

for ai in a:
    ellipsoid = mesher.OblateEllipsoid(0, 0, 0, ai, b0, -13, 10, 180)
    N1, N2 = oblate_ellipsoid.demag_factors(ellipsoid)
    n11_oblate.append(N1)
    n22_oblate.append(N2)

Plot the results



In [7]:

    
x = np.linspace(-1, 1, 11)
xlables = x.copy()
xlables[xlables < 0] *= -1

print x
print xlables









    



[-1.  -0.8 -0.6 -0.4 -0.2  0.   0.2  0.4  0.6  0.8  1. ]
[ 1.   0.8  0.6  0.4  0.2  0.   0.2  0.4  0.6  0.8  1. ]



In [8]:

    
lenght = 3.27
plt.close('all')
plt.figure(figsize=(lenght, 0.6*lenght))

label_fontsize = 10
label_tex_fontsize = 10

plt.plot([0, 0], [0, 1], 'k-')
plt.plot([-1, 1], [0.5, 0.5], 'k-')
plt.plot([-1, 1], [1/3 + 2/15, 1/3 - 2/15], color=(0.6, 0.6, 0.6))
plt.plot([-1, 1], [1/3 - 1/15, 1/3 + 1/15], color=(0.6, 0.6, 0.6))
plt.plot(-e2, n11_oblate, color='r')
plt.plot(-e2, n22_oblate, color='g')
plt.plot(e2, n11_prolate, color='r')
plt.plot(e2, n22_prolate, color='g')
plt.ylabel('demagnetizing factor')
#plt.xlabel('$\epsilon^{2}$', fontsize=label_tex_fontsize)
plt.xlim(-1, 1)
plt.ylim(0, 1)
plt.xticks(x, xlables)
plt.yticks(np.linspace(0, 1, 6))

plt.annotate(s='Oblate', xy=(0.17,0.85), 
              xycoords = 'axes fraction', color='k',
              fontsize=label_fontsize, backgroundcolor='w')
plt.annotate(s='Prolate', xy=(0.66,0.85), 
              xycoords = 'axes fraction', color='k',
              fontsize=label_fontsize, backgroundcolor='w')

plt.annotate(s='$1 - m^{2}$', xy=(0.29,0.02), color='k',
             xycoords='figure fraction', fontsize=label_fontsize)
plt.annotate(s='$(m^{2} - 1)/m^{2}$', xy=(0.64,0.02), color='k',
             xycoords='figure fraction', fontsize=label_fontsize)

plt.grid()

plt.tight_layout()

plt.show()

Demagnetizing factors $\tilde{n}^{\dagger}_{11}$ (in red) and $\tilde{n}^{\dagger}_{22}$ (in green) for prolate (Eqs. 34 and 35) and oblate (Eqs. 36 and 37) ellipsoids as functions of eccentricity $\epsilon$. For prolate ellipsoids, $\epsilon^{2} = (m^{2} - 1)/m^{2}$, and for oblate ellipsoids, $\epsilon^{2} = 1 - m^{2}$, where $m = a/b$. The straight lines of slopes $-2/15$ and $+1/15$ (in grey) are tangential to the curves at $\epsilon^{2} = 0$.

Stoner (1945, Table I)

Data presented by Stoner (1945, Table I)



In [9]:

    
fname = 'Stoner1945_TableI.txt'
file_path = os.path.join(os.path.pardir, 'data', fname)
m, Da = np.loadtxt(file_path, unpack=True)

Triaxial ellipsoids 1



In [10]:

    
# semi-axes (in m)
b0 = 100.1
c0 = 99.9
a = 100*m[m > 1]

# demagnetizing factors
n11_triaxial1 = []
n22_triaxial1 = []
n33_triaxial1 = []

for ai in a:
    ellipsoid = mesher.TriaxialEllipsoid(0, 0, 0, ai, b0, c0, -43, 10, -1.7)
    N1, N2, N3 = triaxial_ellipsoid.demag_factors(ellipsoid)
    n11_triaxial1.append(N1)
    n22_triaxial1.append(N2)
    n33_triaxial1.append(N3)

Prolate ellipsoids



In [11]:

    
# semi-axes (in m)
b0 = 100
a = b0*m[m > 1]

# demagnetizing factors
n11_prolate = []
n22_prolate = []

for ai in a:
    ellipsoid = mesher.ProlateEllipsoid(0, 0, 0, ai, b0, 0, 40, 1)
    N1, N2 = prolate_ellipsoid.demag_factors(ellipsoid)
    n11_prolate.append(N1)
    n22_prolate.append(N2)

Triaxial ellipsoids 2



In [12]:

    
# semi-axes (in m)
a0 = 100.1
b0 = 100
c = b0*m[m < 1]

# demagnetizing factors
n11_triaxial2 = []
n22_triaxial2 = []
n33_triaxial2 = []

for ci in c:
    ellipsoid = mesher.TriaxialEllipsoid(0, 0, 0, a0, b0, ci, -43, 10, -1.7)
    N1, N2, N3 = triaxial_ellipsoid.demag_factors(ellipsoid)
    n11_triaxial2.append(N1)
    n22_triaxial2.append(N2)
    n33_triaxial2.append(N3)

Oblate ellipsoids



In [13]:

    
# semi-axes (in m)
b0 = 100
a = b0*m[m < 1]

# demagnetizing factors
n11_oblate = []
n22_oblate = []

for ai in a:
    ellipsoid = mesher.OblateEllipsoid(0, 0, 0, ai, b0, 15, 100, -7)
    N1, N2 = oblate_ellipsoid.demag_factors(ellipsoid)
    n11_oblate.append(N1)
    n22_oblate.append(N2)

Plot the results



In [21]:

    
lenght = 3.27
plt.close('all')
plt.figure(figsize=(lenght, 4))

label_fontsize = 10
label_tex_fontsize = 10

plt.subplot(2, 1, 1)
plt.semilogy(m[m > 1], Da[m > 1], 'k-', linewidth=2)
plt.semilogy(m[m > 1], n11_prolate, 'r--', linewidth=1.5)
plt.semilogy(m[m > 1], n11_triaxial1, 'b-', linewidth=0.5)
plt.ylabel('$\\tilde{n}^{\dagger}_{11}$')
#plt.xlabel('$m$', fontsize=label_tex_fontsize)
plt.xlim(m[m > 1].min(), m[m > 1].max())
plt.grid()
plt.annotate(s='(a)', xy=(0.92,0.90), 
              xycoords = 'axes fraction', color='k',
              fontsize=label_fontsize)

plt.subplot(2, 1, 2)
plt.plot(m[m < 1], Da[m < 1], 'k-', linewidth=2)
plt.plot(m[m < 1], n11_oblate, 'r--', linewidth=1.5)
plt.plot(m[m < 1], n33_triaxial2, 'b-', linewidth=0.5)
plt.ylabel('$\\tilde{n}^{\dagger}_{11}$')
plt.xlabel('$m$', fontsize=label_tex_fontsize)
plt.xlim(m[m < 1].min(), m[m < 1].max())
plt.grid()
plt.annotate(s='(b)', xy=(0.92,0.90), 
              xycoords = 'axes fraction', color='k',
              fontsize=label_fontsize)

plt.tight_layout()

plt.show()

Comparison between the demagnetizing factors calculated here by using our routines and those presented by Stoner (1945, Table I) (continuous black lines). (a) The blue and red lines represent, respectively, the demagnetizing factor $\tilde{n}^{\dagger}_{11}$ of triaxial ellipsoids (Eq. 29) and the demagnetizing factor $\tilde{n}^{\dagger}_{11}$ of prolate ellipsoids (Eq. 34). (b) The blue and red lines represent, respectively, the demagnetizing factor $\tilde{n}^{\dagger}_{33}$ of triaxial ellipsoids (Eq. 31) and the demagnetizing factor $\tilde{n}^{\dagger}_{11}$ of oblate ellipsoids (Eq. 36).



In [ ]: