µMAG Standard Problem #3 Results
See the problem specification.
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Solution directory
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:
 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_{s}^{2})
 For the partial energies I find:
 Flower  Vortex 
Demagnetization  .2794  .0783 
Exchange  .0177  .1723 
Anisotropy  .0056  .0521 
 Partial magnetization:
 Flower  Vortex 
m_{x}  .000  .000 
m_{y}  .000  .352 
m_{z}  .971  .000 
 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: 493514659537
FAX: 493514659537
email: rave@ifwdresden.de
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 
e_{demag} 
e_{exch} 
e_{anis} 
<m_{z}> 
e_{demag} 
e_{exch} 
e_{anis} 
<m_{x}> 
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 
From: Riccardo Hertel
and Helmut Kronmüller
MaxPlanckInstitut für Metallforschung
Postfach 80 06 65, 70569 Stuttgart, Germany
Date: 22 Jul 1998
Flower state  Vortex state
 Critical edge length (in units of l_{ex}): L=8.52,
mean energy density in reduced units: e=0.3049.
 Partial energy densities:
 Flower  Vortex 
Demag.  0.2839  0.0830 
Exch.  0.0158  0.1696 
Anis.  0.0052  0.0522 
 Mean reduced magnetization:
Flower:  <m_{z}>=0.973 
Vortex:  <m_{y}>=0.351 
Twisted Flower State
However, according to our calculation the symmetric flower state
is metastable at this edge length (L=8.52).
The twisted flower state is found to be a singledomain state of lower
energy compared with the symmetric flower state.
 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.
 Partial energy densities:
 Twisted Flower  Vortex 
Demag.  0.2332  0.0821 
Exch.  0.0466  0.1689 
Anis.  0.0233  0.0521 
 Mean reduced magnetization:
Twisted Flower:  <m_{z}>=0.874 
Vortex:  <m_{y}>=0.344 
Images:
 Flower state
 Twisted flower state
 Vortex state
Please send comments
to rmcmichael@nist.gov and
join the µMAG discussion email list.
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23JUL1998