Week 2

Pair distribution function
- Example1
Temperature and g(r)

Second week (Feb 14th-18th): Data analysis

The computational simulation are very useful, because you can obtain different properties, like physical, chemical, etc., of the material in study. The simulations are like virtual experiments, you can change the conditions to which a material is subjected as temperature, pressure, etc. With different conditions you can obtain different responses of the material, this means different properties. In the left figure you can see a box with atoms (in three different times), in the right part of the box there is a piston, we give the piston a fixed velocity during a little time. So the atoms start to move in the same direction that the piston moves, when the piston is stopped the atoms continue moving in that direction, this is known as a shock wave.

One of the properties that we can see and evaluate is the change in the density, take a look to the graph, and explain what it is means. Well, there are others interesting properties that you can evaluate:

Thermodynamic Properties: Temperature, Pressure, etc
Structural Properties: Pair Distribution Function g(r), Angle distribution, Common neighbor analysis CNA, Coordination number, etc.

This time we will see the Pair Distribution Function.

Pair distribution function

The pair–distribution function g(r) is defined in such a way that, sitting on one atom, the probability of finding another one atom in a spherical shell between r and r + dr is <n(r,r + dr)> = rho 4 pi 2 r² g(r) dr, where rho=N/V is the density (N: number of atoms, V: volume). The next figure show very clear what means the pair distribution function.

In the left of the figure you can see the atoms are neatly distributed. The first neighbor of any atom is at a distance of 2 Angstrom. See the central atom, the red line show who are the first neighbors, there are four and are at 2 Angstrom of the central atom. If you do the same for all atoms you will found that the distance of the first neighbors is 2 A always!, but the number of the first neighbors varies, for example, see what happens for the atoms in the corners. Now take a look to the graph, the x-axis is distance in A, and y-axis is the g(r). The first peak is at 2 A and it tell us that are a number of atoms that are at 2 A of distance, this first peak indicates the distance of the first neighbors. You can do the same analysis for the second (yellow line), third (light blue line), etc, neighbors.

Example 1

Create a week2 folder inside of your "hannah" folder. Enter to the week2 folder, copy the exampleweek2.tar.gz from /home/lab/claudial/, you can go to week1 for remember how create a folder and copy and unzip a file.

When you unzip the file, you will find two folders: gdr and temperature, go to the gdr folder.

In gdr folder you will find 2 files: argon.control and Fe.control. Run the LPMD program for each one!!

argon.control is a simulation with 108 argon atoms at 0 Kelvin so is a solid with face centered cubic struture (fcc) (3D) and the value of lattice constant is a=5.26 A. We want to see the g(r) property.

 $lpmd argon.control

Fe.control do a configuration with 31,250 iron atoms, is a solid with body center cubic structure (bcc) (3d) and the value of lattice constant is a=2.87 A. We want to see the g(r) property.

 $lpmd-analyzer Fe2.control

More information of Crystal structure here!

After ran the simulation you will see these files gdrArgon.dat and gdrFe.dat with the information about the pair distribution function! Graph each one and attach here:

Calculate the values for the first, second and third neighbor for the fcc and bcc structures using the lattice parameters for Ar and Fe. Compare the values with graphics!!

In the comment part (at the end of the page), write the values obtained by theory, and make some comments about the values obtained by g(r).

Temperature and g(r)

Now you will compare the g(r) for the same system, but with different temperatures.

Go to the temperature folder:

 $cd /home/lab/hannah/week2/temperature

you will see the argon50K.control file, look it. What is the temperature?, now run it!

 $nohup lpmd argon50K.control &

Do the temperature and gdr graphics, and upload them!

Compare the g(r) obtained before at 0K with this new one! make conclusion in the comment part ;)

Now copy the "argon50K.control" file with different name:

 $cp argon50K.control argon200K.control

Edit argon200K.control file, and change the number of steps to 10000, temperature of the system to 200 K, and the output file names. Where? here!

 $gedit argon200K.control

and change

 output module=xyz file=argon.xyz -> argon200K.xyz
 steps 5000 -> 10000
 .
 .
 prepare temperature t=50.0 -> 200.0
 monitor step,temperature,... output=datos.dat -> datos200K.dat
 output gdrArgon.dat -> gdrArgon200.dat

run the simulation and see what happens with the g(r) graph! uppload it :)

Make comments about the general results! ;)

hannah ? — 18 February 2011, 16:10

1. My calculated values for the neighbors of Ar and Fe are: Ar- first neighbor: 3.72

    second neighbor: 5.62
    third neighbor: 7.44

Fe- first neighbor: 2.45

    second neighbor: 2.87
    third neighbor: 4.059

These values are from what I can tell, very similar to the values plotted by the g(r) function. The only number that doesnt match up is the value I got for the third neighbor of argon. This is probably wrong because my picture was very bad and I couldn't determine which atom would be the third neighbor, so my distance is probably to an incorrect atom.

2. The g(r) obtained at 50K for Ar was similar to the graph from 0K, however the peaks on the 50K graph have much more area underneath them. The peaks seem to be in the same places, it's the area that is the main difference. In the 0K simulation the atoms dont have velocity, so there are distinct vertical peaks where there are atoms and there is nothing on the graph in between peaks indicating the "empty space" between atoms. This area under the peaks on the graph of 50K is probably due to the fact that in the 50K simulation, the atoms have velocity and therefore are moving. There is less "empty space" between atoms because the atoms are constantly moving, both closer and farther away from each other.

3. When the simulation was run at 200K the graph again has similar peak points, but there is even more area under the curve. From this we can discern that there is no "empty space" between atoms, and this makes sense given that Claudia told me that at this temperature Argon is a liquid. In a liquid the atoms are constantly sliding and moving past each other, they are no longer fixed in the face centered cube.

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