Magnet

In mrfmsim, we assume the magnets are:

• uniformly magnetized

• magnetized in the \(z\)-direction

Currently, the two types of magnets supported are

mrfmsim.component.magnet.SphereMagnet(...)	Spherical magnet object with its Bz, Bzx, Bzxx calculations.
mrfmsim.component.magnet.RectangularMagnet(...)	Rectangular magnet object with the bz, bzx, bzxx calculations.
mrfmsim.component.cylindermagnet.CylinderMagnetApprox(...)	Cylinder magnet object approximated by Rectangular Magnets.

Example Usage

To create a nickel spherical magnet (\(\mu_0 M_s = 1800 \: \mathrm{mT}\)) with a radius of \(r = 50 \: \mathrm{nm}\):

from mrfmsim.component import SphereMagnet
magnet = SphereMagnet(magnet_radius=50.0, mu0_Ms=800.0, magnet_origin=[0.0, 0.0, 0.0])

To display the magnet information:

>>> print(magnet)
SphereMagnet
  magnet_radius = 50.0 nm
  magnet_origin = [0.0, 0.0, 0.0] nm
  mu0_Ms = 800.000 mT

magnet module

class mrfmsim.component.magnet.RectangularMagnet(magnet_length: list[float], magnet_origin: list[float], mu0_Ms: float)

Bases: ComponentBase

Rectangular magnet object with the bz, bzx, bzxx calculations.

Parameters:

• magnet_length (list) – length of rectangular magnet in \((x, y, z)\) direction [nm]

• magnet_origin (list) – the position of the magnet origin \((x, y, z)\) [nm]

• mu0_Ms (float) – saturation magnetization [mT]

Bz_method(x, y, z)

Calculate magnetic field \(B_z\) [mT].

The magnetic field is calculated following Ravaud2009 [1].

The magnet is set up so that the \(x\) and \(y\) dimensions are centered about the zero point. The translation in \(z\) shifts the tip of the magnet in the \(z\)-direction to be the given distance from the surface.

Using the Coulombian model, assuming a uniform magnetization throughout the volume of the magnet and modeling each face of the magnet as a layer of continuous current density. The total field is found by summing over the faces.

The magnetic field is given by:

\[B_z = \dfrac{\mu_0 M_s}{4\pi} \sum_{i=1}^{2} \sum_{j=1}^2 \sum_{k=1}^2(-1)^{i+j+k} arctan \left( \dfrac{(x - x_i)(y - y_i))}{(z - z_k)R} \right)\]

Here \((x,y,z)\) are the coordinates for the location at which we want to know the field; The magnet spans from x1 to x2 in the \(x\)-direction, y1 to y2 in the \(y\)-direction, and z1 to z2 in the \(z\)-direction;

\[R = \sqrt{(x - x_i)^2 + (y - y_j)^2 + (z - z_k)^2}\]

where \(\mu_0 M_s\) is the magnet’s saturation magnetization in mT.

[1]

Ravaud, R. and Lemarquand, G. “Magnetic field produced by a parallelepipedic magnet of various and uniform polarization” , PIER, 2009, 98, 207-219 [10.2528/PIER09091704].

Parameters:

• x (float) – \(x\) coordinate of sample grid [nm]

• y (float) – \(y\) coordinate of sample grid [nm]

• z (float) – \(z\) coordinate of sample grid [nm]

Bzx_method(x, y, z)

Calculate magnetic field gradient \(B_{zx}\).

\(B_{zx} \equiv \partial B_z / \partial x\) [\(\mathrm{mT} \: \mathrm{nm}^{-1}\)]. The magnetic field gradient \(B_{zx} = \dfrac{\partial{B_z}}{\partial x}\) is given by the following:

\[B_{zx} = \dfrac{\mu_0 M_s}{4 \pi} \sum_{i=1}^2 \sum_{j=1}^2 \sum_{k=1}^2(-1)^{i+j+k} \left( \dfrac{(y-y_j)(z-z_k)}{ R((x-x_i)^2 + (z-z_k)^2))} \right)\]

As described above, \((x,y,z)\) are coordinates for the location at which we want to know the field gradient; the magnet spans from x1 to x2 in the x-direction, y1 to y2 in the y-direction, and from z1 to z2 in the z-direction;

\[R = \sqrt{(x - x_i)^2 + (y - y_j)^2 + (z - z_k)^2}\]

\(\mu_0 M_s\) is the magnet’s saturation magnetization in mT.

Parameters:

• x (float) – \(x\) coordinate [nm]

• y (float) – \(y\) coordinate [nm]

• z (float) – \(z\) coordinate [nm]

Bzxx_method(x, y, z)

Calculate magnetic field second derivative \(B_{zxx}\).

\(B_{zxx} \equiv \partial^2 B_z / \partial x^2\) [ \(\mathrm{mT} \; \mathrm{nm}^{-2}\)] The magnetic field’s second derivative is given by the following:

\[B_{zxx} = \dfrac{\partial^2 B_z}{\partial x^2} = \dfrac{\mu_0 M_s}{4 \pi} \sum_{i=1}^2 \sum_{j=1}^2 \sum_{k=1}^2(-1)^{i+j+k} \left( \dfrac{-(x-x_i)(y-y_j)(z-z_k) (3(x-x_i)^2 +2(y-y_j)^2 + 3(z-z_k)^2)} {((x-x_i)^2 + (y-y_j)^2 + (z-z_k)^2)^{3/2} ((x-x_i)^2 + (z-z_k)^2)^2} \right)\]

with the variables defined above.

Parameters:

• x (float) – \(x\) coordinate [nm]

• y (float) – \(y\) coordinate [nm]

• z (float) – \(z\) coordinate [nm]

magnet_length: list[float]

magnet_origin: list[float]

mu0_Ms: float

class mrfmsim.component.magnet.SphereMagnet(magnet_radius: float, magnet_origin: list[float], mu0_Ms: float)

Bases: ComponentBase

Spherical magnet object with its Bz, Bzx, Bzxx calculations.

Parameters:

• magnet_radius (float) – sphere magnet radius [nm]

• magnet_origin (list) – the position of the magnet origin \((x, y, z)\) [nm]

• mu0_Ms (float) – saturation magnetization [mT]

Bz_method(x, y, z)

Calculate magnetic field \(B_z\) [mT].

Parameters:

• x (float) – \(x\) coordinate of sample grid [nm]

• y (float) – \(y\) coordinate of sample grid [nm]

• z (float) – \(z\) coordinate of sample grid [nm]

The magnetic field is calculated as

\[B_z = \dfrac{\mu_0 M_s}{3} \left( 3 \dfrac{Z^2}{R^5} - \dfrac{1}{R^3} \right) R = \sqrt{X^2+Y^2+Z^2}\]

Here \((x,y,z)\) is the location at which we want to know the field; \((x_0, y_0, z_0)\) is the location of the center of the magnet; \(r\) is the radius of the magnet; \(X = (x-x_0)/r\); \(Y = (y-y_0)/r\), and \(Z = (z-z_0)/r\) are normalized distances to the center of the magnet; \(\mu_0 M_s\) is the magnetic sphere’s saturation magnetization in mT.

Bzx_method(x, y, z)

Calculate magnetic field gradient \(B_{zx}\).

\(B_{zx} \equiv \partial B_z / \partial x\) [ \(\mathrm{mT} \: \mathrm{nm}^{-1}\) ]. With \(X\), \(Y\), \(Z\), \(R\), \(r\), and \(\mu_0 M_s\) defined in Bz(x, y, z), the magnetic field gradient is calculated as

\[B_{zx} = \dfrac{\partial B_z}{\partial x} = \dfrac{\mu_0 M_s}{r} X \: \left( \dfrac{1}{R^5} - 5 \dfrac{Z^2}{R^7} \right) R = \sqrt{X^2+Y^2+Z^2}\]

Parameters:

• x (float) – \(x\) coordinate of sample grid [nm]

• y (float) – \(y\) coordinate of sample grid [nm]

• z (float) – \(z\) coordinate of sample grid [nm]

Returns:

magnetic field gradient

Return type:

np.array

Bzxx_method(x, y, z)

Calculate magnetic field second derivative \(B_{zxx}\).

\(B_{zxx} \equiv \partial^2 B_z / \partial x^2\) [\(\mathrm{mT} \: \mathrm{nm}^{-2}\)]. The inputs are With \(X\), \(Y\), \(Z\), \(R\), \(r\), and \(\mu_0 M_s\) defined as above, the magnetic field second derivative is calculated as

\[B_{zxx} = \dfrac{\partial^2 B_z}{\partial x^2} = \dfrac{\mu_0 M_s}{r^2} \: \left( \dfrac{1}{R^5} - 5 \dfrac{X^2}{R^7} - 5 \dfrac{Z^2}{R^7} + 35 \dfrac{X^2 Z^2}{R^9} \right) R = \sqrt{X^2+Y^2+Z^2}\]

Parameters:

• x (float) – \(x\) coordinate of sample grid [nm]

• y (float) – \(y\) coordinate of sample grid [nm]

• z (float) – \(z\) coordinate of sample grid [nm]

Returns:

magnetic field second derivative

Return type:

np.array

magnet_origin: list[float]

magnet_radius: float

mu0_Ms: float

cylindermagnet module

class mrfmsim.component.cylindermagnet.CylinderMagnetApprox(magnet_radius: float, magnet_length: float, magnet_origin: tuple[float, float, float], mu0_Ms: float)

Bases: ComponentBase

Cylinder magnet object approximated by Rectangular Magnets.

Parameters:

• magnet_radius (float) – cylinder magnet radius [nm]

• magnet_length (float) – cylinder magnet length [nm]

• magnet_origin (tuple) – the position of the magnet origin \((x, y, z)\) [nm]

• mu0_Ms (float) – saturation magnetization [mT]

Bz_method(x, y, z)

Calculate magnetic field \(B_z\) [mT].

Approximating Cylinder Magnet by 11 Rectangular Magnets. When viewed from the vertical direction, we are using a row of rectangles to approximate a circle, these rectangular blocks are arranged side by side.

The magnetic field of each rectangular magnet is calculated following the method described in Ravaud2009 [2]. The magnet is set up so that the \(x\) and \(y\) dimensions are centered about the zero point. The translation in \(z\) shifts the tip of the magnet in the \(z\)-direction to be the given distance from the surface. Using the Coulombian model, assuming a uniform magnetization throughout the volume of the magnet and modeling each face of the magnet as a layer of continuous current density. The total field is found by summing over the faces. The magnetic field is given by:

\[B_z = \dfrac{\mu_0 M_s}{4\pi} \sum_{i=1}^{2} \sum_{j=1}^2 \sum_{k=1}^2(-1)^{i+j+k} \arctan \left( \dfrac{(x - x_i)(y - y_i))}{(z - z_k)R} \right)\]

Here \((x,y,z)\) are the coordinates for the location at which we want to know the field; The magnet spans from \(x_1\) to \(x_2\) in the \(x\)-direction, \(y_1\) to \(y_2\) in the \(y\)-direction, and \(z_1\) to \(z_2\) in the \(z\)-direction;

\[R = \sqrt{(x - x_i)^2 + (y - y_j)^2 + (z - z_k)^2}\]

where \(\mu_0 M_s\) is the magnet’s saturation magnetization in mT.

[2]

Ravaud, R. and Lemarquand, G. “Magnetic field produced by a parallelepipedic magnet of various and uniform polarization”, PIER, 2009, 98, 207-219 [10.2528/PIER09091704].

Parameters:

• x (float) – \(x\) coordinate of sample grid [nm]

• y (float) – \(y\) coordinate of sample grid [nm]

• z (float) – \(z\) coordinate of sample grid [nm]

Bzx_method(x, y, z)

Calculate magnetic field gradient \(B_{zx}\).

Approximating Cylinder Magnet by 11 Rectangular Magnets. When viewed from the vertical direction, we are using a row of rectangles to approximate a circle, these rectangular blocks are arranged side by side.

The magnetic field gradient for RectangularMagnet is: \(B_{zx} = \dfrac{\partial{B_z}}{\partial x}\) is given by the following:

\[B_{zx} = \dfrac{\mu_0 M_s}{4 \pi} \sum_{i=1}^2 \sum_{j=1}^2 \sum_{k=1}^2(-1)^{i+j+k} \left( \dfrac{(y-y_j)(z-z_k)}{ R((x-x_i)^2 + (z-z_k)^2))} \right)\]

As described above, \((x,y,z)\) are coordinates for the location at which we want to know the field gradient; the magnet spans from x1 to x2 in the \(x\)-direction, y1 to y2 in the \(y\)-direction, and from z1 to z2 in the \(z\)-direction;

\[R = \sqrt{(x - x_i)^2 + (y - y_j)^2 + (z - z_k)^2}\]

\(\mu_0 M_s\) is the magnet’s saturation magnetization in mT.

Parameters:

• x (float) – \(x\) coordinate [nm]

• y (float) – \(y\) coordinate [nm]

• z (float) – \(z\) coordinate [nm]

Bzxx_method(x, y, z)

Calculate magnetic field second derivative \(B_{zxx}\).

Approximating Cylinder Magnet by 11 Rectangular Magnets. When viewed from the vertical direction, we are using a row of rectangles to approximate a circle, these rectangular blocks are arranged side by side.

The magnetic field second derivative for RectangularMagnet is: \(B_{zxx} \equiv \partial^2 B_z / \partial x^2\) [ \(\mathrm{mT} \; \mathrm{nm}^{-2}\)] The magnetic field’s second derivative is given by the following:

\[B_{zxx} = \dfrac{\partial^2 B_z}{\partial x^2} = \dfrac{\mu_0 M_s}{4 \pi} \sum_{i=1}^2 \sum_{j=1}^2 \sum_{k=1}^2(-1)^{i+j+k} \left( \dfrac{-(x-x_i)(y-y_j)(z-z_k) (3(x-x_i)^2 +2(y-y_j)^2 + 3(z-z_k)^2)} {((x-x_i)^2 + (y-y_j)^2 + (z-z_k)^2)^{3/2} ((x-x_i)^2 + (z-z_k)^2)^2} \right)\]

with the variables defined above.

Parameters:

• x (float) – \(x\) coordinate [nm]

• y (float) – \(y\) coordinate [nm]

• z (float) – \(z\) coordinate [nm]

magnet_length: float

magnet_origin: tuple[float, float, float]

magnet_radius: float

mu0_Ms: float