µMAG Standard Problem #3 Results

See the problem specification.
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Solution directory


Submitted Solution

Date: 13 Mar 1998
From: Wolfgang Rave

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 lex for Q=0.1 is L = 8.47, total reduced energy e=0.3027 (in units of Km=2 piMs2)
  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:
The method of calculation is an extension of the methods developed by Berkov, Ramstoeck and Hubert to 3D. More details can be found in our forthcoming paper "The possible magnetic states of small cubic particles with uniaxial anisotropy".

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

Date: Wed, 24 Jun 1998
From: Jose L. Martins

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.
 

 
Flower
Vortex
N
L
e
edemag
eexch
eanis
<mz>
edemag
eexch
eanis
<mx>
10
8.4073
0.2937
0.2699
0.0180
0.0057
0.9703
0.0687
0.1736 0.0513
0.3385
20
8.4528
0.3001
0.2766
0.0178
0.0057
 0.9707
0.0754
0.1727
0.0519
0.3482
30
8.4635
0.3014
0.2780
0.0178
0.0056
0.9709
0.0769
0.1725
0.0520
0.3500
40
8.4654
0.3020
0.2787
0.0177
0.0056
0.9709
0.0775
0.1724
0.0521
0.3508
50
8.4667
0.3023
0.2789
0.0177
0.0056
0.9709
0.0778
0.1724
0.0521
0.3511
70
8.4673
0.3025
0.2792
0.0177
0.0056
0.9710
0.0780
0.1724
0.0521
0.3513
infinity
8.4687 0.3026 0.2792 0.0177 0.0056 0.9710 0.0780 0.1724 0.0521 0.3516
 
 

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 lex): 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

However, according to our calculation the symmetric flower state is meta-stable at this edge length (L=8.52). The twisted flower state is found to be a single-domain state of lower energy compared with the symmetric flower state.
  1. The critical edge length for the transition Twisted Flower state -- Vortex state is L=8.57·lex, 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:


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23-JUL-1998