µMAG Standard Problem #3 Results

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Solution directory

• 13 Mar 1998 Wolfgang Rave, Alex Hubert and Karl Fabian
• 24 Jun 1998 Filipe Ribeiro, Paulo Freitas, and José Luís Martins
• 22 Jul 1998 Riccardo Hertel and Helmut Kronmüller
• 08 Sep 2021 Rasmus Bjørk, E. B. Poulsen and A. R. Insinga

Submitted Solution

I am writing on behalf of Alex Hubert, Karl Fabian and myself to submit our solution to your standard problem #3:

1. The single domain limit in units of l_ex for Q=0.1 is L = 8.47, total reduced energy e=0.3027 (in units of K_m=2 piM_s²)
2. For the partial energies I find:

   	Flower	Vortex
   Demagnetization	.2794	.0783
   Exchange	.0177	.1723
   Anisotropy	.0056	.0521

3. Partial magnetization:

   	Flower	Vortex
   mx	.000	.000
   my	.000	.352
   mz	.971	.000

4. Images:
   • Flower state (44.6 kB)
   • Vortex state (40.7 kB)

Yours sincerely,
Wolfgang Rave

------------------------------------------------------------------
IFW Dresden
Helmholtzstr. 20
01069 Dresden
Germany
phone: 49-351-4659537
FAX:   49-351-4659537
e-mail: rave@ifw-dresden.de

Submitted Solution

Our group at INESC, Lisbon, Portugal (Filipe Ribeiro, Paulo Freitas, and José Luís Martins) has performed detailed simulations on the µMAG standard problem #3. Our results are very close to those recently submitted by Hubert, Fabian and Rave.

We used a cubic grid NxNxN and found that the calculated properties were almost linear when plotted in a 1/N**2 scale. We give below the results for N=10, 20, 30, 40, 50, 70 and the results of an extrapolation to an infinite grid.
 

Submitted Solution

From: Riccardo Hertel and Helmut Kronmüller
Max-Planck-Institut für Metallforschung
Postfach 80 06 65, 70569 Stuttgart, Germany
Date: 22 Jul 1998

Flower state -- Vortex state

1. Critical edge length (in units of l_ex): L=8.52, mean energy density in reduced units: e=0.3049.

2. Partial energy densities:

   	Flower	Vortex
   Demag.	0.2839	0.0830
   Exch.	0.0158	0.1696
   Anis.	0.0052	0.0522

3. Mean reduced magnetization:

   Flower:	<mz>=0.973
   Vortex:	<my>=0.351

Twisted Flower State

1. The critical edge length for the transition Twisted Flower state -- Vortex state is L=8.57·l_ex, and the mean energy density in reduced units in this case is e=0.3032.
2. Partial energy densities:

   	Twisted Flower	Vortex
   Demag.	0.2332	0.0821
   Exch.	0.0466	0.1689
   Anis.	0.0233	0.0521

3. Mean reduced magnetization:

   Twisted Flower:	<mz>=0.874
   Vortex:	<my>=0.344

Images:

• Flower state
  • 3D view(10.2k)
• Twisted flower state
  • 3D view(10.0k),
  • x-sections(8.8k) and
  • z-sections(6.5k).
• Vortex state
  • 3D view(9.2k),
  • x-sections(8.6k) and
  • z-section(4.4k).

Submitted Solution

Reference: J. Magn. Magn. Mater., .v 535, pp. 168057 (2021), DOI: https://doi.org/10.1016/j.jmmm.2021.168057

We used the MagTense open source micromagnetic simulation framework to compute solutions to standard problem #3. We used a cubic Cartesian mesh of dimensions N×N×N to compute the crossover energy between the flower and the vortex state. The energy is determined with a precision of 1e-4 using an fminsearch approach.

We find the results as tabulated below. Extrapolating the energy crossover to an infinitely fine resolution by fitting a power law the data gives L_∞ = 8.477 ± 0.007.