.. _example 2D periodicity:

Example: 2D periodicity
-----------------------

This example is another application of the macro-geometry feature,
where we now deal with a 2D "thin film" system. The unit cell is a
``30x10x10 nm^3`` prism 

.. image:: /example_2Dperiodicity/mesh.png
   :align: center
   :width: 559
   :height: 290

where we take 10 copies in x- and 40 copies in y-direction to create
the macro geometry.

The script :download:`no_periodic.py </example_2Dperiodicity/no_periodic.py>` simulates behaviour of
just the unit cell of size 30x10x10 nm^3 (without any periodic
copies):

.. include:: /example_2Dperiodicity/no_periodic.py
  :literal:

and the relaxation curves are obtained via:

.. include:: /example_2Dperiodicity/no_periodic.gnu
  :literal:

which creates the following plot:

.. image:: /example_2Dperiodicity/no_periodic.png
   :align: center
   :width: 750
   :height: 525

From this plot we can see that with using only the unit cell the
magnetisation aligns along the x-axis at equilibrium.

We now move to the macro geometry of a thin film with dimensions 400x300x10nm^3 which is realised in :download:`periodic2.py </example_2Dperiodicity/periodic2.py>`.

.. comment:

  Employing periodicity means to break down the number of degrees of
  freedom in a complex system by introducing the extra constraint that
  magnetisation will be the same within all cells. 
  In principle, we have
  some freedom in the choice of cell whose behaviour will be
  replicated. While this most often will be the central cell of our
  sample, there are some situations where one may want to place this in
  a different position. For the sake of providing an example, we put our
  unit cell on the (-1,-1, 0) point of the lattice as shown in
  :download:`periodic2.py </example_2Dperiodicity/periodic2.py>` and let the
  system relax.

  
.. include:: /example_2Dperiodicity/periodic2.py
  :literal:

As in the previous example, we first define the three unit vectors of
the lattice, again slightly larger than the dimension of the unit cell
to avoid overlapping (and thus to eleminate any exchange coupling
across the interfaces for this demonstration of the demagnetisation effects)::

   x_lattice = 30.01  # the spacing is 0.01
   y_lattice = 10.01  # the spacing is 0.01
   z_lattice = 0.0

Then we define where the copies will be placed::

    for xi in range(-4,6):
        for yi in range(-19,21):
            lattice_points.append([xi*x_lattice,yi*y_lattice,0.0*z_lattice])
    
    # copies of the system along the x-axis
    pbc = nmag.SetLatticePoints(vectorlist=lattice_points, scalefactor=SI(1e-9,'m'))

The simulation cell is (always) the one at the (0,0,0) lattice
point. The for loops therefore place 4 copies of the simulation cell
in the negative x direction [i.e. (-4,0,0), (-3,0,0), (-2,0,0), and
(-1,0,0)] and 5 in the positive the x direction [i.e. (1,0,0),
(2,0,0), (3,0,0), (4,0,0), (5,0,0)]. The translation vector (0,0,0)
corresponds to the actual simulation cell.

Similarly, the inner for loop places 20 copies along the positive
y-axis and 19 along the negative one.

We set the same initial configuration as before, with a uniform
magnetisation along [1,1,1], and let the system evolve towards the
equilibrium.

The outcome is shown in the following figure:

.. image:: /example_2Dperiodicity/periodic2.png
   :align: center
   :width: 750
   :height: 525

where we notice that the final configuration is now with the
magnetisation aligned along the (negative) y axis, and not along the x
axis as before. The alignment along the y-direction is expected, as
now the macro geometry has a total size of 300.09 nm times 400.39 nm
(30 nm x 10 copies plus spacings along the x direction times 10 nm x
40 copies plus spacings along the y direction) times 10nm (no periodic
copies along the z direction), so the longest side now is along the y
direction. The demagnetisation energy of the macro geometry drives the
alignment of the magnetisation with the y-direction.

Other usage examples include this study [#BordignonEtAl]_ of an array of interacting triangular rings.

.. [#BordignonEtAl] Giuliano Bordignon, Thomas Fischbacher, Matteo Franchin, Jurgen P. Zimmermann, Peter A. J. de Groot, Hans Fangohr, *Numerical studies of demagnetizing effects in triangular ring arrays*, Journal of Applied Physics **103** 07D932 (2008), online at http://eprints.soton.ac.uk/50995/