.. _Example: Uniaxial anisotropy:

Example: Uniaxial anisotropy
----------------------------

In this example we would like to simulate the development of a Bloch type
domain wall on a thin cobalt bar of dimension 504 x 1 x 1 nm
(:download:`bar.nmesh.h5 </example_uniaxial_anis/bar.nmesh.h5>`) due to uniaxial
anisotropy.

.. _Uniaxial anisotropy simulation script:

Uniaxial anisotropy simulation script
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. include:: bar.py
  :literal:

We shall now discuss the :download:`bar.py </example_uniaxial_anis/bar.py>`  script 
step-by-step:

In this particular example we are solely interested in energy terms resulting
from exchange interaction and anisotropy. Hence we disable the demagnetisation
field as follows::

  sim = nmag.Simulation(do_demag=False)

We then create the material ``Co`` used for the bar, cobalt in this case, which exhibits :ref:`uniaxial_anisotropy` in z direction with phenomenological anisotropy constant
``K1 = SI(520e3, "J/m^3")``::

  Co = nmag.MagMaterial(name="Co",
                        Ms=SI(1400e3, "A/m"),
                        exchange_coupling=SI(30e-12, "J/m"),
                        anisotropy=nmag.uniaxial_anisotropy(axis=[0, 0, 1], K1=SI(520e3, "J/m^3")))

After loading the mesh, we set the initial magnetisation direction such that
it rotates from +z to -z while staying in the plane normal to x direction
(hence suggesting the development of a Bloch type domain wall)::

  def sample_m0((x, y, z)):
    # relative_position goes linearly from -1 to +1 in region B
    relative_position = -2*(x - offset)/length + 1
    mz = min(1.0, max(-1.0, relative_position))
    return [0, math.sqrt(1 - mz*mz), mz]

We further pin the magnetisation at the very left (x < offset = 2 nm)
and right (x > offset + length = 502 nm) of the bar
(note that the pinning function may also just return a python truth
value rather than the number 0.0 or 1.0)::

  def sample_pinning((x, y, z)):
    return x >= offset and x <= offset + length

  sim.set_pinning(sample_pinning)

Finally, we relax the system to find the equilibrium magnetisation
configuration, which is saved to the file ``bar_mag_x.dat`` in a format
understandable by `Gnuplot`_.

.. _Visualization:

Visualization
~~~~~~~~~~~~~

We can then use the following `Gnuplot`_ script to visualize
the equilibrium magnetisation:

.. include:: bar_mag_x.gnu
  :literal:

The resulting plot clearly shows that a Bloch type domain wall has developed:

.. image:: /example_uniaxial_anis/bar_mag_x.png
  :align: center
  :width: 800
  :height: 600

The figure shows also that the Bloch domain wall is well localized at the center
of the bar, in the region where x goes from 200 to 300 nm.

.. _Comparison:

Comparison
~~~~~~~~~~

After simulating the same scenario with
OOMMF (see :download:`oommf/bar.mif </example_uniaxial_anis/oommf/bar.mif>`),
we can compare results using another `Gnuplot`_ script:

.. include:: bar_mag_x_compared.gnu
  :literal:

which generates the following plot showing good agreement of both systems:

.. image:: /example_uniaxial_anis/bar_mag_x_compared.png
  :align: center
  :width: 800
  :height: 600

The plot shows also the known analytical solution::

  Mz(x) = Ms * cos(pi/2 + atan(sinh((x - x_wall)/sqrt(A/K1))))

The plot shows only a restricted region located at the center of the bar,
thus allowing an easier comparison between the three sets of data.