forked from lijiext/lammps
Updated doc page for triclinic cell
git-svn-id: svn://svn.icms.temple.edu/lammps-ro/trunk@8004 f3b2605a-c512-4ea7-a41b-209d697bcdaa
This commit is contained in:
athomps
parent
0686ff8955
commit
44d79b97ca
Binary file not shown.
|
After Width: | Height: | Size: 18 KiB |
|
|
@ -0,0 +1,26 @@
|
|||
\documentclass[12pt]{article}
|
||||
|
||||
\usepackage{amsmath}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\begin{eqnarray*}
|
||||
\mathbf{x}
|
||||
&=&
|
||||
\begin{pmatrix}
|
||||
\mathbf{a} &
|
||||
\mathbf{b} &
|
||||
\mathbf{c}
|
||||
\end{pmatrix}
|
||||
\cdot
|
||||
\frac{1}{V}
|
||||
\begin{pmatrix}
|
||||
\mathbf{B \times C} \\
|
||||
\mathbf{C \times A} \\
|
||||
\mathbf{A \times B}
|
||||
\end{pmatrix}
|
||||
\cdot
|
||||
\mathbf{X}
|
||||
\end{eqnarray*}
|
||||
|
||||
\end{document}
|
||||
Binary file not shown.
|
Before Width: | Height: | Size: 24 KiB After Width: | Height: | Size: 58 KiB |
|
|
@ -1,17 +1,27 @@
|
|||
\documentclass[12pt]{article}
|
||||
|
||||
\usepackage{amsmath}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\begin{eqnarray*}
|
||||
\vec{a} &=& (a_x,0,0) \\
|
||||
\vec{b} &=& (b_x,b_y,0) \\
|
||||
\vec{c} &=& (c_x,c_y,c_z) \\
|
||||
\begin{pmatrix}
|
||||
\mathbf{a} &
|
||||
\mathbf{b} &
|
||||
\mathbf{c}
|
||||
\end{pmatrix}
|
||||
& = &
|
||||
\begin{pmatrix}
|
||||
a_x & b_x & c_x \\
|
||||
0 & b_y & c_y \\
|
||||
0 & 0 & c_z \\
|
||||
\end{pmatrix} \\
|
||||
a_x &=& A \\
|
||||
b_x &=& (\vec{B} \bullet \vec{A}) \,\, / \,\, A \\
|
||||
b_y &=& |\vec{A} \times \vec{B}| \,\, / \,\, A \quad \rm{or} \quad \sqrt{B^2 - {b_x}^2} \\
|
||||
c_x &=& (\vec{C} \bullet \vec{A}) \,\, / \,\, A \\
|
||||
c_y &=& [\vec{C} \bullet ((\vec{A} \times \vec{B}) \times \vec{A})] \,\, / \,\, |(\vec{A} \times \vec{B}) \times \vec{A}| \quad \rm{or} \quad \sqrt{C^2 - {c_x}^2 -{c_z}^2} \\
|
||||
c_z &=& [\vec{C} \bullet (\vec{A} \times \vec{B})] \,\, / \,\, |\vec{A} \times \vec{B}| \\
|
||||
b_x &=& \mathbf{B} \cdot \mathbf{\hat{A}} \quad = \quad B \cos{\gamma} \\
|
||||
b_y &=& |\mathbf{\hat{A}} \times \mathbf{B}| \quad = \quad B \sin{\gamma} \quad = \quad \sqrt{B^2 - {b_x}^2} \\
|
||||
c_x &=& \mathbf{C} \cdot \mathbf{\hat{A}} \quad = \quad C \cos{\beta} \\
|
||||
c_y &=& \mathbf{C} \cdot \widehat{(\mathbf{A} \times \mathbf{B})} \times \mathbf{\hat{A}} \quad = \quad \frac{B C - b_x c_x}{b_y} \\
|
||||
c_z &=& |\mathbf{C} \cdot \widehat{(\mathbf{A} \times \mathbf{B})}|\quad = \quad \sqrt{C^2 - {c_x}^2 - {c_y}^2} \\
|
||||
\end{eqnarray*}
|
||||
|
||||
\end{document}
|
||||
|
|
|
|||
|
|
@ -769,11 +769,11 @@ See the <A HREF = "dump.html">dump</A> command for more information on XTC files
|
|||
</H4>
|
||||
<P>By default, LAMMPS uses an orthogonal simulation box to encompass the
|
||||
particles. The <A HREF = "boundary.html">boundary</A> command sets the boundary
|
||||
conditions of the box (periodic, non-,periodic, etc). The orthogonal
|
||||
conditions of the box (periodic, non-periodic, etc). The orthogonal
|
||||
box has its "origin" at (xlo,ylo,zlo) and is defined by 3 edge vectors
|
||||
starting from the origin given by <B>a</B> = (xhi-xlo,0,0); <B>b</B> =
|
||||
(0,yhi-ylo,0); <B>c</B> = (0,0,zhi-zlo). The 6 parameters
|
||||
(xlo,xhi,ylo,yhi,zlo,zhi) are defined at the time the simluation box
|
||||
(xlo,xhi,ylo,yhi,zlo,zhi) are defined at the time the simulation box
|
||||
is created, e.g. by the <A HREF = "create_box.html">create_box</A> or
|
||||
<A HREF = "read_data.html">read_data</A> or <A HREF = "read_restart.html">read_restart</A>
|
||||
commands. Additionally, LAMMPS defines box size parameters lx,ly,lz
|
||||
|
|
@ -781,30 +781,53 @@ where lx = xhi-xlo, and similarly in the y and z dimensions. The 6
|
|||
parameters, as well as lx,ly,lz, can be output via the <A HREF = "thermo_style.html">thermo_style
|
||||
custom</A> command.
|
||||
</P>
|
||||
<P>LAMMPS also allows simulations to be perfored in triclinic
|
||||
<P>LAMMPS also allows simulations to be performed in triclinic
|
||||
(non-orthogonal) simulation boxes shaped as a parallelepiped with
|
||||
triclinic symmetry. The parallelepiped has its "origin" at
|
||||
(xlo,ylo,zlo) and is defined by 3 edge vectors starting from the
|
||||
origin given by <B>a</B> = (xhi-xlo,0,0); <B>b</B> = (xy,yhi-ylo,0); <B>c</B> =
|
||||
(xz,yz,zhi-zlo). <I>Xy,xz,yz</I> can be 0.0 or positive or negative values
|
||||
(xz,yz,zhi-zlo). <I>xy,xz,yz</I> can be 0.0 or positive or negative values
|
||||
and are called "tilt factors" because they are the amount of
|
||||
displacement applied to faces of an originally orthogonal box to
|
||||
transform it into the parallelepiped. Note that in LAMMPS the
|
||||
transform it into the parallelepiped. In LAMMPS the
|
||||
triclinic simulation box edge vectors <B>a</B>, <B>b</B>, and <B>c</B> cannot be
|
||||
arbitrary vectors. As indicated, <B>a</B> must be aligned with the x axis,
|
||||
<B>b</B> must be in the xy plane, and <B>c</B> is arbitrary. However, this is
|
||||
not a restriction since it is possible to rotate any set of 3 crystal
|
||||
basis vectors so that they meet this restriction.
|
||||
arbitrary vectors. As indicated, <B>a</B> must lie on the positive x axis.
|
||||
<B>b</B> must lie in the xy plane, with strictly positive y component. <B>c</B> may
|
||||
have any orientation with strictly positive z component.
|
||||
The requirement that <B>a</B>, <B>b</B>, and <B>c</B> have strictly positive x, y,
|
||||
and z components, respectively, ensures
|
||||
that <B>a</B>, <B>b</B>, and <B>c</B> form a complete right-handed basis.
|
||||
These restrictions impose
|
||||
no loss of generality, since it is possible to rotate/invert
|
||||
any set of 3 crystal basis vectors so that they conform to the restrictions.
|
||||
</P>
|
||||
<P>For example, assume that the 3 vectors <B>A</B>,<B>B</B>,<B>C</B> are the edge
|
||||
vectors of a general parallelipied, where there is no directional
|
||||
requirements <B>A</B>,<B>B</B>,<B>C</B> other than they are not co-planar and that
|
||||
<B>C</B> dotted into (<B>A</B> x <B>B</B>) be > 0, i.e. the vectors are ordered to
|
||||
satisfy a right-hand rule. The equivalent LAMMPS <B>a</B>,<B>b</B>,<B>c</B> vectors
|
||||
can be computed as follows where A = |<B>A</B>| = scalar length of <B>A</B>.
|
||||
vectors of a general parallelepiped, where there is no restriction
|
||||
on <B>A</B>,<B>B</B>,<B>C</B> other than they form a complete right-handed basis i.e.
|
||||
<B>A</B> x <B>B</B> . <B>C</B> > 0.
|
||||
The equivalent LAMMPS <B>a</B>,<B>b</B>,<B>c</B> are a linear rotation of <B>A</B>, <B>B</B>, and
|
||||
<B>C</B> and can be computed as follows:
|
||||
</P>
|
||||
<CENTER><IMG SRC = "Eqs/transform.jpg">
|
||||
</CENTER>
|
||||
<P>where A = |<B>A</B>| indicates the scalar length of <B>A</B>. The ^ hat symbol
|
||||
indicates the corresponding unit vector. beta and gamma are angles
|
||||
between the vectors described below. The same rotation must also
|
||||
be applied to atom positions, velocities, and any other vector quantities.
|
||||
This can be done by first converting to fractional coordinates in the
|
||||
old basis and then converting to distance coordinates in the new basis.
|
||||
The transformation is given by the following equation:
|
||||
</P>
|
||||
<CENTER><IMG SRC = "Eqs/rotate.jpg">
|
||||
</CENTER>
|
||||
<P>where V is the volume of the box, <B>X</B> is the original vector quantity and
|
||||
<B>x</B> is the vector in the LAMMPS basis.
|
||||
</P>
|
||||
<P>If it should happen that
|
||||
<B>A</B>, <B>B</B>, and <B>C</B> form a left-handed basis, then it is necessary
|
||||
to first apply an inversion in addition to rotation. This can be achieved
|
||||
by interchanging two of the vectors or changing the sign of one of them.
|
||||
</P>
|
||||
<P>There is no requirement that a triclinic box be periodic in any
|
||||
dimension, though it typically should be in at least the 2nd dimension
|
||||
of the tilt (y in xy) if you want to enforce a shift in periodic
|
||||
|
|
@ -1214,7 +1237,7 @@ discussed below, it can be referenced via the following bracket
|
|||
notation, where ID in this case is the ID of a compute. The leading
|
||||
"c_" would be replaced by "f_" for a fix, or "v_" for a variable:
|
||||
</P>
|
||||
<DIV ALIGN=center><TABLE BORDER=1 >
|
||||
<DIV ALIGN=center><TABLE WIDTH="0%" BORDER=1 >
|
||||
<TR><TD >c_ID </TD><TD > entire scalar, vector, or array</TD></TR>
|
||||
<TR><TD >c_ID[I] </TD><TD > one element of vector, one column of array</TD></TR>
|
||||
<TR><TD >c_ID[I][J] </TD><TD > one element of array
|
||||
|
|
@ -1413,7 +1436,7 @@ data and scalar/vector/array data.
|
|||
input, that could be an element of a vector or array. Likewise a
|
||||
vector input could be a column of an array.
|
||||
</P>
|
||||
<DIV ALIGN=center><TABLE BORDER=1 >
|
||||
<DIV ALIGN=center><TABLE WIDTH="0%" BORDER=1 >
|
||||
<TR><TD >Command</TD><TD > Input</TD><TD > Output</TD><TD ></TD></TR>
|
||||
<TR><TD ><A HREF = "thermo_style.html">thermo_style custom</A></TD><TD > global scalars</TD><TD > screen, log file</TD><TD ></TD></TR>
|
||||
<TR><TD ><A HREF = "dump.html">dump custom</A></TD><TD > per-atom vectors</TD><TD > dump file</TD><TD ></TD></TR>
|
||||
|
|
|
|||
|
|
@ -760,11 +760,11 @@ See the "dump"_dump.html command for more information on XTC files.
|
|||
|
||||
By default, LAMMPS uses an orthogonal simulation box to encompass the
|
||||
particles. The "boundary"_boundary.html command sets the boundary
|
||||
conditions of the box (periodic, non-,periodic, etc). The orthogonal
|
||||
conditions of the box (periodic, non-periodic, etc). The orthogonal
|
||||
box has its "origin" at (xlo,ylo,zlo) and is defined by 3 edge vectors
|
||||
starting from the origin given by [a] = (xhi-xlo,0,0); [b] =
|
||||
(0,yhi-ylo,0); [c] = (0,0,zhi-zlo). The 6 parameters
|
||||
(xlo,xhi,ylo,yhi,zlo,zhi) are defined at the time the simluation box
|
||||
(xlo,xhi,ylo,yhi,zlo,zhi) are defined at the time the simulation box
|
||||
is created, e.g. by the "create_box"_create_box.html or
|
||||
"read_data"_read_data.html or "read_restart"_read_restart.html
|
||||
commands. Additionally, LAMMPS defines box size parameters lx,ly,lz
|
||||
|
|
@ -772,30 +772,53 @@ where lx = xhi-xlo, and similarly in the y and z dimensions. The 6
|
|||
parameters, as well as lx,ly,lz, can be output via the "thermo_style
|
||||
custom"_thermo_style.html command.
|
||||
|
||||
LAMMPS also allows simulations to be perfored in triclinic
|
||||
LAMMPS also allows simulations to be performed in triclinic
|
||||
(non-orthogonal) simulation boxes shaped as a parallelepiped with
|
||||
triclinic symmetry. The parallelepiped has its "origin" at
|
||||
(xlo,ylo,zlo) and is defined by 3 edge vectors starting from the
|
||||
origin given by [a] = (xhi-xlo,0,0); [b] = (xy,yhi-ylo,0); [c] =
|
||||
(xz,yz,zhi-zlo). {Xy,xz,yz} can be 0.0 or positive or negative values
|
||||
(xz,yz,zhi-zlo). {xy,xz,yz} can be 0.0 or positive or negative values
|
||||
and are called "tilt factors" because they are the amount of
|
||||
displacement applied to faces of an originally orthogonal box to
|
||||
transform it into the parallelepiped. Note that in LAMMPS the
|
||||
transform it into the parallelepiped. In LAMMPS the
|
||||
triclinic simulation box edge vectors [a], [b], and [c] cannot be
|
||||
arbitrary vectors. As indicated, [a] must be aligned with the x axis,
|
||||
[b] must be in the xy plane, and [c] is arbitrary. However, this is
|
||||
not a restriction since it is possible to rotate any set of 3 crystal
|
||||
basis vectors so that they meet this restriction.
|
||||
arbitrary vectors. As indicated, [a] must lie on the positive x axis.
|
||||
[b] must lie in the xy plane, with strictly positive y component. [c] may
|
||||
have any orientation with strictly positive z component.
|
||||
The requirement that [a], [b], and [c] have strictly positive x, y,
|
||||
and z components, respectively, ensures
|
||||
that [a], [b], and [c] form a complete right-handed basis.
|
||||
These restrictions impose
|
||||
no loss of generality, since it is possible to rotate/invert
|
||||
any set of 3 crystal basis vectors so that they conform to the restrictions.
|
||||
|
||||
For example, assume that the 3 vectors [A],[B],[C] are the edge
|
||||
vectors of a general parallelipied, where there is no directional
|
||||
requirements [A],[B],[C] other than they are not co-planar and that
|
||||
[C] dotted into ([A] x [B]) be > 0, i.e. the vectors are ordered to
|
||||
satisfy a right-hand rule. The equivalent LAMMPS [a],[b],[c] vectors
|
||||
can be computed as follows where A = |[A]| = scalar length of [A].
|
||||
vectors of a general parallelepiped, where there is no restriction
|
||||
on [A],[B],[C] other than they form a complete right-handed basis i.e.
|
||||
[A] x [B] . [C] > 0.
|
||||
The equivalent LAMMPS [a],[b],[c] are a linear rotation of [A], [B], and
|
||||
[C] and can be computed as follows:
|
||||
|
||||
:c,image(Eqs/transform.jpg)
|
||||
|
||||
where A = |[A]| indicates the scalar length of [A]. The ^ hat symbol
|
||||
indicates the corresponding unit vector. beta and gamma are angles
|
||||
between the vectors described below. The same rotation must also
|
||||
be applied to atom positions, velocities, and any other vector quantities.
|
||||
This can be done by first converting to fractional coordinates in the
|
||||
old basis and then converting to distance coordinates in the new basis.
|
||||
The transformation is given by the following equation:
|
||||
|
||||
:c,image(Eqs/rotate.jpg)
|
||||
|
||||
where V is the volume of the box, [X] is the original vector quantity and
|
||||
[x] is the vector in the LAMMPS basis.
|
||||
|
||||
If it should happen that
|
||||
[A], [B], and [C] form a left-handed basis, then it is necessary
|
||||
to first apply an inversion in addition to rotation. This can be achieved
|
||||
by interchanging two of the basis vectors or changing the sign of one of them.
|
||||
|
||||
There is no requirement that a triclinic box be periodic in any
|
||||
dimension, though it typically should be in at least the 2nd dimension
|
||||
of the tilt (y in xy) if you want to enforce a shift in periodic
|
||||
|
|
|
|||
Loading…
Reference in New Issue