Numerical integration of a magnetic spin vector in a magnetic field

[chaiyar]

Homework Statement

Hi there, thanks in advance for any help!

I have a first order DE: [tex] \frac{\partial \vec{m}}{\partial t} = -\vec{m} \times \vec{h}_{eff} + \alpha \vec{m} \times \frac{\partial \vec{m}}{\partial t}[/tex] (a scaled Landau-Lifshitz-Gilbert equation)

where m is a magnetism vector, alpha is a damping factor and h is an effective uniform magnetic field.

I'm trying to numerically integrate it with the Euler method to get a precession of the spin vector around the h vector.

So far I've integrated the first term but the second, damping term I can't see how to translate into code.. (in C++)

So essentially what I've done is assumed the magnetic field to be aligned in the z-direction, and written

for(i=1 ; i<=tmax ; i++) {

					mx = mx + h * -my;
					my = my + h *  mx;
					t = t + h;

for the first term, where mx is the x-component of the m vector etc. and h is the timestep, and I need to add the damping term onto the end.

Considering the x-component first, presumably the derivative in the damping term has to be put down as m_x as well, but what direction is the other m vector to be taken in?

Sorry if that's not very well asked, but thanks a lot for any help!

 

[TSny]

Hello, chaiyar.

If you write out the DE for each component (x, y, z), you will get three equations that are linear in ##\dot{m_x}, \dot{m_y}, \dot{m_z}##, where the dot denotes time derivative.

Algebraically solving these simultaneously for ##\dot{m_x}, \dot{m_y}, \dot{m_z}##, you can get equations of the form

##\dot{m_x} = f_1(m_x, m_y, m_z)##
##\dot{m_y} = f_2(m_x, m_y, m_z)##
##\dot{m_z} = f_3(m_x, m_y, m_z)##

for certain functions ##f_1, f_2, f_3## .

Then you can use these equations with the Euler method to step forward in time.

 

[chaiyar]

Thanks for your help TSny, but I'm still a little lost.

I end up with three equations of the form

[tex]

\dot{m}_x = f_1 (m_y,m_z,\dot{m}_y,\dot{m}_z) \\
\dot{m}_y = f_2 (m_x,m_z,\dot{m}_x,\dot{m}_z) \\
\dot{m}_z = f_3 (m_x,m_y,\dot{m}_x,\dot{m}_y) \\

[/tex]

but I don't see how to incorporate the time derivatives into the program.. Do I have to take the equations further to get rid of the derivatives by using Lagrange multipliers or something like that?

Thanks again!

 

[voko]

You have three equations that are linear in ## \dot{m}_i ##, so you always can convert them to ## \dot{m}_i = f(...) ##, where the right hand side is independent of ## \dot{m}_i ##.

 

[paranormal]

I would like to first commend you on your efforts to numerically integrate the first order DE for the magnetic spin vector. This is a complex and important problem in the field of magnetism and your approach is a good starting point.

In order to incorporate the damping term into your numerical integration, you will need to consider the direction of the magnetism vector in both the cross product and the derivative. The damping term essentially represents a torque acting on the spin vector, causing it to align with the effective magnetic field. Therefore, the direction of the damping term will depend on the orientation of the spin vector itself.

To incorporate this into your code, you will need to use the cross product between the spin vector and the derivative of the spin vector, as well as the cross product between the spin vector and the effective magnetic field. This will give you the direction of the torque acting on the spin vector, which you can then use to update the components of the spin vector in your numerical integration.

I would also recommend considering different numerical integration methods, such as the Runge-Kutta method, which may provide more accurate results for this type of problem.

Overall, I appreciate your approach and would encourage you to continue exploring this problem and seeking guidance from experts in the field. Good luck with your research!