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6.1.2 Technical Matters
- What units?
- What about boundaries?
- What initial confiuration?
- How to adjust density and temperature
Units:
To avoid the handling of small numbers, choose units appropriate to
the model.
Lennard-Jones:
Energy unit
; length .
The pair energy is then
where
and
.
As the third mechanical unit, choose the atomic mass
.
The time unit is now the combination
.
Electrical charge: best measured in multiples of the
electron charge,
.
Number density: is a large number; therefore we
reduce it by a suitable standard density,
:
.
Temperature:
Hard spheres:
no ``natural'' unit of energy; therefore choose
self-consistent time unit
.
For hard spheres of diameter
the customary
standard density is
; thus:
.
Periodic boundary conditions (PBC) and nearest image
convention (NIC):
To avoid ``wall effects''we surround the basic cell containing
the particles by periodic images of itself.
Spin lattices:
In each coordinate direction the last (``rightmost'') spin in a row
interacts also with a right neighbor, which is taken to be identical to
the first (leftmost) spin in that row, and vice versa (see Figure).
Figure 6.1:
Periodic boundary conditions on a spin lattice
|
Fluid: Apply the following rule:
For each particle store, instead of any coordinate ,
the quantity
(with the side length of the cell).
When a particle leaves the cell to the right,
it is automatically replaced by a particle entering from the left, etc.
Nearest Image Convention:
In computing pair vectors between two particles
and one needs only differences of coordinate values.
If this coordinate difference
is larger than , then
the particle will not be regarded as an interaction partner of ;
instead, its left periodic image with coordinate interacts
with .
The NIC rule may be implemented by a sequence of IF commands or
by the more compact code line
where nint() denotes the rounded value of , i.e. the integer nearest
to . This formulation of NIC is also better suited for vectorising
compilers.
Starting configuration:
On an Ising lattice, draw spin values with equal probabilities
for and .
Molecules in disordered media:
Overlaps must be avoided.
Place the molecules on a lattice, then ``melt''this crystal before
the actual simulation run: Thermalization.
Population number in a cubic cell with face-centered cubic arrangement:
, with . Therefore typical particle numbers in
simulations are
etc.
Adjusting density and temperature:
Given , the density is adjusted to a desired value by shrinking or expanding
the volume: scale all coordinates by a suitable factor.
The temperature is a constant parameter in an MC simulation.
In molecular dynamics it must be adjusted in the following manner.
Since
(with
) or, in reduced units,
we first take the average of
over a
number of MD steps to determine the actual temperature of the simulated
system. Then we scale each velocity component according to
Since is a fluctuating quantity it can be adjusted
only approximately.
EXERCISE:
Reduced units:
Consider a pair of Lennard-Jones particles with
and
(typical for Argon).
Let the two molecules be situated at a distance of
from each other.
- Calculate the potential energy of this arrangement.
- Do the same calculation using and as units of
energy and length, respectively.
These parameters then vanish from the expression for the pair energy,
and the calculation is done with quantities of order .
- With the above units for energy and length, together with the
atomic mass unit, compute the metric value of the self-consistent unit
of time? Let one of the particles have a metric speed , typical
of the thermal velocities of atoms or small molecules. What is the value
of in self-consistent units?
PROJECT MC/MD:
As a first reusable module for a simulation program, write a code to
set up a cubic box of side length inhabited by up to particles
in a face-centered cubic arrangement. Use your favourite programming language
and make the code flexible enough to allow for easy change of volume
(i.e. density). Make sure that the lengths are measured in units
of . For later reference, let us call this subroutine
STARTCONF.
Advice: It is convenient to count the lower, left and front
face of the cube as belonging to the basic cell, while the three other
faces belong to the next periodic cells.
To test for correct arragement of the particles, compute the diagnostic
which is sometimes called ``melting factor''. For a fcc
configuration it should be equal to (why?).
By scaling all lengths, adjust the
volume such that the reduced number density becomes .
PROJECT MD:
Augment the subroutine STARTCONF by a procedure that assigns random
velocities to the particles, making sure that the
sum total of each velocity component is zero.
PROJECT MC/MD:
The second subroutine will serve to compute the total potential
energy in the system, assuming a Lennard-Jones interaction and applying
the nearest image convention:
Write such a subroutine and call it ENERGY. Use it to compute the
energy in the system created by STARTCONF.
Next: 6.2 Monte Carlo Method
Up: 6.1 Model Systems of
Previous: 6.1.1 Fluids and solids
Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001