Magnet#

Summary#

Here we are assuming the magnets are:

  • uniformly magnetized

  • magnetized in the z direction

Currently the two types of magnets supported

mrfmsim_marohn.component.magnet.SphereMagnet(...)

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

mrfmsim_marohn.component.magnet.RectangularMagnet(...)

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

Example Usage#

Create a radius \(r = 50 \: \mathrm{nm}\) sphere of cobalt (\(\mu_0 M_s = 1800 \: \mathrm{mT}\))

magnet = SphereMagnet(radius=50., mu0_Ms=1800., origin=[0., 0., 0.])

Print the magnet information:

print(magnet)

magnet module#

class mrfmsim_marohn.component.magnet.RectangularMagnet(length, mu0_Ms, origin)[source]#

Bases: mrfmsim.component.ComponentBase

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

Bz_method(x, y, z)[source]#

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

The magnetic field is calculated following Ravaud2009. 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.

Reference: 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].

  • set the magnet up so that the x and y dimensions are centered about the zero point.

  • The translation in z should shift the tip of the magnet in the z-direction to be the given distance from the surface.

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)[source]#

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;

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

\(\mu_0 M_s\) 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)[source]#

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:

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

with the variables defined above. :param float x: x coordinate [nm] :param float y: y coordinate [nm] :param float z: z coordinate [nm]

class mrfmsim_marohn.component.magnet.SphereMagnet(radius, mu0_Ms, origin)[source]#

Bases: mrfmsim.component.ComponentBase

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

Bz_method(x, y, z)[source]#

Calculate magnetic field \(B_z\) [mT]. :param float x: x coordinate of sample grid [nm] :param float y: y coordinate of sample grid [nm] :param float z: 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)[source]#

Calcualte 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

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

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

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