Calculators
This is a small collection of calculators useful for running β-NMR/β-NQR experiments at TRIUMF (see also https://bnmr.triumf.ca/).
Some of the calculators are web forms, where you simply enter a new value in one of the boxes on the page and press ENTER or click outside the box to update the values. Others are Python scripts, which can be downloaded and run on your own machine.
Table of contents
- Bandwidth-to-pulse-duration converter for modulated RF fields
- β-NMR antenna-pickup-to-RF-field converter
- β-NQR magnetic field calibration and calculator
- Demagnetization factors for common sample geometries
- Beamline optics
Bandwidth-to-pulse-duration converter for modulated RF fields
β-NMR antenna-pickup-to-RF-field converter
The above expression is accurate to ± 3 %. Note that this expression is only valid for data collected before September 2019!
β-NQR magnetic field calibration and calculator
The calibration constants are: a = 0.175 ± 0.046 G and b = 2.2131 ± 0.0019 G / A.
Demagnetization factors for common sample geometries
Ellipsoid
#!/usr/bin/python3
import numpy as np
from scipy.special import ellipeinc
# Demagnetization factor N/4π for an ellipsoid with semi-axes a, b, and c.
#
# J. A. Osborn.
# "Demagnetizing factors of the general ellipsoid".
# Phys. Rev. 67, 351 (1945).
# https://doi.org/10.1103/PhysRev.67.351
#
# General ellipsoid
# Equation (2.3) [a >= b >= c >= 0]
def N_ellipsoid(a, b, c):
# Equation (2.4) - amplitude
theta = np.arccos(c / a)
# Equation (2.5)
phi = np.arccos(b / a)
# Equation (2.6) - modulus
k = np.sin(phi) / np.sin(theta)
alpha = np.arcsin(k)
# convert to the notation used in SciPy
# https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.ellipeinc.html
m = np.power(k, 2)
E = ellipeinc(theta, m)
# Equation (2.3) - demagnetization factor for the general ellipsoid
return (
(np.cos(phi) * np.cos(theta))
/ (np.power(np.sin(theta), 3) * np.power(np.cos(alpha), 2))
* (((np.sin(theta) * np.cos(phi)) / np.cos(theta)) - E)
)Download demagnetization_factor_ellipsoid.py.
Rectangular prism
#!/usr/bin/python3
import numpy as np
# Demagnetization factor N/4π for a rectangular prism with semi-axes a, b, and c.
#
# A. Aharoni.
# "Demagnetizing factors for rectangular ferromagnetic prisms".
# J. Appl. Phys. 83, 3422 (1998).
# https://doi.org/10.1063/1.367113
#
# Eq. (1) - see Fig. 1 for_abc coordinate system
# see also the online version of calculator
# http://www.magpar.net/static/magpar/doc/html/demagcalc.html
def N_prism(a, b, c):
# the expression takes input as half of the semi-axes
a = 0.5 * a
b = 0.5 * b
c = 0.5 * c
# define some convenience terms
a2 = a * a
b2 = b * b
c2 = c * c
abc = a * b * c
ab = a * b
ac = a * c
bc = b * c
r_abc = np.sqrt(a2 + b2 + c2)
r_ab = np.sqrt(a2 + b2)
r_bc = np.sqrt(b2 + c2)
r_ac = np.sqrt(a2 + c2)
# compute the factor
pi_N = (
((b2 - c2) / (2 * bc)) * np.log((r_abc - a) / (r_abc + a))
+ ((a2 - c2) / (2 * ac)) * np.log((r_abc - b) / (r_abc + b))
+ (b / (2 * c)) * np.log((r_ab + a) / (r_ab - a))
+ (a / (2 * c)) * np.log((r_ab + b) / (r_ab - b))
+ (c / (2 * a)) * np.log((r_bc - b) / (r_bc + b))
+ (c / (2 * b)) * np.log((r_ac - a) / (r_ac + a))
+ 2 * np.arctan2(ab, c * r_abc)
+ (a2 * a + b2 * b - 2 * c2 * c) / (3 * abc)
+ ((a2 + b2 - 2 * c2) / (3 * abc)) * r_abc
+ (c / ab) * (r_ac + r_bc)
- (r_ab * r_ab * r_ab + r_bc * r_bc * r_bc + r_ac * r_ac * r_ac) / (3 * abc)
)
# divide out the factor of pi
return pi_N / np.piDownload demagnetization_factor_prism.py.