Draw schematically the radial distribution function $g(r)$ for a Lennard-Jones fluid at low and high particle densities. Discuss in both cases the behavior at small $r$. At high densities $g(r)$ has some damped oscillations. Explain their origin and what would happen at long distances.

**What I know:**

The radial distribution function $g(r)$ describes how density varies as a function of distance from a reference particle (*Wikipedia*).

Specifically in Lennard-Jones case, we have :

Here we can observe that if repulsion outweighs attraction, the curve will die out on the x axis. On the contrary, if it is the other way around, the curve will grow exponentially. For the sake of clarity, I would remark y = 0 as the Ideal Gas behaviour.

I have read that $g(r)$ vanishes at short distances, because the probability of finding two particles close to each other vanishes due to the repulsive part of the potential. At high densities $g(r)$ can show some damped oscillations:

These oscillations express the preference of the particles to be found at specific distances from a reference particle at the origin. For instance in the LJ case, a first layer of particles will be localized closed to the minimum of LJ's potential, which is the origin of the first peak in $g(r)$.

This layer prevents other particles from getting close to it, which is what causes the first

minimum in $g(r)$.

**What I do not know.**

1)The explanation:

'These oscillations express the preference of the particles to be found at specific distances from a reference particle at the origin'.

**How can particles have 'preference' to be at different distances?** I think this is not a good physical explanation. I have been thinking about this and I would say that as there is a high density of particles at small distances, the repulsion force is exerted on these, which triggers such oscillations. Finally there is a point where they are so close to each other that the repulsion force provides them with an initial kinetic energy which will be equal to a final electrostatic potential energy of zero (selecting our zero of electrostatic potential energy at y=0) , and that is the moment when the curve dies out on the x axis. Do you agree with this explanation?**

2)Are we dealing with underdamped oscillations at high densities? Could you give an insight into these kind of oscillations (if what I have just said is not enough or incorrect)?

3)When we are dealing with long distances, would $g(r)$ tend to be one? What would we observe, the Ideal Gas behaviour? Why? Is it because there is no interaction between the particles?

As always, I am interested in explaining such phenomena from a physical point of view.

**ADDITIONAL INFORMATION RELATED TO THE RADIAL DISTRIBUTION:**

I came across the following slide:

Which compares the structure factor $S(Q)$ with the radial distribution function $g(r)$. I know that the structure factor is a mathematical description of how a material **scatters** incident radiation, but **how is the structure factor related to the distribution function (as this slide suggests)?**

edited Dec 11, 2018 by sammy gerbil