fix up typesetting of docs, spelling, and integrated into fix command table

* :doc:`oneway <fix_oneway>` - constrain particles on move in one direction

* :doc:`orient/bcc <fix_orient>` - add grain boundary migration force for BCC

* :doc:`orient/fcc <fix_orient>` - add grain boundary migration force for FCC

* :doc:`orient/eco <fix_orient_eco>` - add generalized grain boundary migration force

* :doc:`phonon <fix_phonon>` - calculate dynamical matrix from MD simulations

* :doc:`pimd <fix_pimd>` - Feynman path integral molecular dynamics

* :doc:`planeforce <fix_planeforce>` - constrain atoms to move in a plane

.. parsed-literal::

   fix ID group-ID orient/eco u0 eta rcut orientationsFile

   fix ID group-ID orient/eco u0 eta cutoff orientationsFile

* ID, group-ID are documented in fix command

* u0 = energy added to each atom (energy units)

* eta = cutoff value (usually 0.25)

* rcut = cutoff radius for orientation parameter calculation 

* cutoff = cutoff radius for orientation parameter calculation

* orientationsFile = file that specifies orientation of each grain

Examples

of atoms to either of the two grains forming the grain boundary is

determined from an orientation-dependent order parameter as described

in :ref:`(Ulomek) <Ulomek>`. The potential energy of atoms is either increased by an amount

of 0.5*u0 or -0.5*u0 according to the orientation of the surrounding 

of 0.5*\ *u0* or -0.5*\ *u0* according to the orientation of the surrounding

crystal. This creates a potential energy gradient which pushes atoms near

the grain boundary to orient according to the energetically favorable

grain orientation. This fix is designed for applications in bicrystal system

.. math::

\begin{eqnarray}

w(|\vec{r}_{jk}|)=w_{jk}=\left\{\begin{array}{lc}

\frac{|\vec{r}_{jk}|^{4}}{r_{\mathrm{cut}}^{4}}-2\frac{|\vec{r}_{jk}|^{2}}{r_{\mathrm{cut}}^{2}}+1, & |\vec{r}_{jk}|<r_{\mathrm{cut}} \\

   w(|\vec{r}_{jk}|) = w_{jk} & = \left\{\begin{array}{lc} \frac{|\vec{r}_{jk}|^{4}}{r_{\mathrm{cut}}^{4}}

     -2\frac{|\vec{r}_{jk}|^{2}}{r_{\mathrm{cut}}^{2}}+1, & |\vec{r}_{jk}|<r_{\mathrm{cut}} \\

      0, & |\vec{r}_{jk}|\ge r_{\mathrm{cut}}

\end{array}\right.

\label{eq:envelope}

\end{eqnarray}

\begin{eqnarray}

\chi_{j} & = & \frac{1}{N}\sum_{l=1}^{3}\left\lbrack\left\vert\psi_{l}^{\mathrm{I}}(\vec{r}_{j})\right\vert^{2}-\left\vert\psi_{l}^{\mathrm{II}}(\vec{r}_{j})\right\vert^{2}\right\rbrack

\label{eq:force-long}

\end{eqnarray}

\begin{eqnarray}

\psi_{l}^{\mathrm{X}}(\vec{r}_{j})=\sum_{k\in\mathit{\Gamma}_{j}}w_{jk}\exp\left(\mathrm{i}\vec{r}_{jk}\cdot\vec{q}_{l}^{\mathrm{X}}\right)

\end{eqnarray}

\begin{eqnarray}

u(\chi_{j}) & = & \frac{u_{0}}{2}\left\{\begin{array}{lc}

      \end{array}\right. \\

   \chi_{j} & = \frac{1}{N}\sum_{l=1}^{3}\left\lbrack\left\vert\psi_{l}^{\mathrm{I}}(\vec{r}_{j})\right\vert^{2}-\left\vert\psi_{l}^{\mathrm{II}}(\vec{r}_{j})\right\vert^{2}\right\rbrack \\

   \psi_{l}^{\mathrm{X}}(\vec{r}_{j}) & = \sum_{k\in\mathit{\Gamma}_{j}}w_{jk}\exp\left(\mathrm{i}\vec{r}_{jk}\cdot\vec{q}_{l}^{\mathrm{X}}\right) \\

   u(\chi_{j}) & = \frac{u_{0}}{2}\left\{\begin{array}{lc}

   1, & \chi_{j}\ge\eta\\

   \sin\left(\frac{\pi\chi_{j}}{2\eta}\right), &  -\eta<\chi_{j}<\eta\\

   -1, & \chi_{j}\le-\eta

   \end{array}\right.

\label{eq:energy-mid}

\end{eqnarray}

which are fully explained in :ref:`(Ulomek) <Ulomek>`

and :ref:`(Schratt & Mohles) <Schratt>`.

The force on each atom is the negative gradient of the synthetic potential energy. It

depends on the surrounding of this atom. An atom far from the grain boundary does not

experience a synthetic force as its surrounding is that of an oriented single crystal

and thermal fluctuations are masked by the parameter eta. Near the grain boundary 

and thermal fluctuations are masked by the parameter *eta*\ . Near the grain boundary

however, the gradient is nonzero and synthetic force terms are computed.

The orientationsFile specifies the perfect oriented crystal basis vectors for the

two adjoining crystals. The first three lines (line=row vector) for the energetically penalized and the

last three lines for the energetically favored grain assuming u0 is positive. For 

negative u0, this is reversed. With the rcut parameter, the size of the region around 

each atom which is used in the order parameter computation is defined. rcut must be 

last three lines for the energetically favored grain assuming *u0* is positive. For

negative *u0*, this is reversed. With the *cutoff* parameter, the size of the region around

each atom which is used in the order parameter computation is defined. The cutoff must be

smaller than the interaction range of the MD potential. It should at

least include the nearest neighbor shell. For high temperatures or low angle

grain boundaries, it might be beneficial to increase rcut in order to get a more 

grain boundaries, it might be beneficial to increase the cutoff in order to get a more

precise identification of the atoms surrounding. However, computation time will

increase as more atoms are considered in the order parameter and force computation.

It is also worth noting that the cutoff radius must not exceed the communication

contains the three components of a primitive lattice vector oriented according to

the grain orientation in the simulation box. The first (last) three lines correspond

to the primitive lattice vectors of the first (second) grain. An example for

a :math:`\Sigma\langle001\rangle` misorientation is given at the end.

a :math:`\Sigma\langle001\rangle` mis-orientation is given at the end.

If no synthetic energy difference between the grains is created, u0=0, the 

If no synthetic energy difference between the grains is created, :math:`u0=0`, the

force computation is omitted. In this case, still, the order parameter of the

driving force is computed and can be used to track the grain boundary motion throughout the

simulation.

For illustration purposes, here is an example file that specifies a

Sigma=5 <001> tilt grain boundary. 

This is for a lattice constant of 3.52 Angs.

:math:`\Sigma=5 \langle 001 \rangle` tilt grain boundary. This is for a lattice constant of 3.52 Angstrom:

.. parsed-literal::

    sigma5.ori:

.. parsed-literal::

    1.671685  0.557228  1.76212

    0.557228 -1.671685  1.76212

    2.228913 -1.114456 0.000000

    2.228913 -1.114456  0.00000

    0.557228  1.671685  1.76212

    1.671685 -0.557228  1.76212

    2.228913 1.114456 0.000000

    2.228913  1.114456  0.00000

bgq

Bh

Bialke

bicrystal

Biersack

bigbig

bigint

Branduardi

Branicio

brennan

Brien

Brilliantov

Broadwell

Broglie

ebt

ec

Ec

eco

ecoul

ecp

Ecut

mofff

MOFFF

Mohd

Mohles

mol

Mol

molfile

orangered

organometallic

orientational

orientationsFile

orientorder

Orlikowski

ornl

Schiotz

Schlitter

Schmid

Schratt

Schoen

Schotte

Schulten

ulb

Uleft

uloop

Ulomek

ulsph

uMech

umin