# Correlation length of surface roughness

1. Jan 22, 2013

### orzyszpon

I need a simple mental physical picture of interface roughness. If the auto-correlation function of surface roughness is Gaussian, does a long correlation length connote a smooth surface or not? I do not seem to be able to find a simple picture that would appeal to me physically. Thanks all for helping.

2. Feb 5, 2013

### russel0110019

Hello,

I had been reading few papers recently and came across the term "surface correlation length". Can you please let me know what what does it physically mean?

Bests.

3. Feb 8, 2013

### M Quack

The surface roughness is usually described by a RMS (root mean square) value of the height fluctuations. Large RMS means rough, small means smooth.

However, RMS does not tell you about the length scale parallel to the surface.

If you describe this by a Gaussian auto-correlation function, then the width of the Gaussian will tell you about the length scales.

Think of your surface as composed of square blocks of different height.
Pick a point on one of the blocks, then pick a second point.

If the second point is on the same block, it will be at the same height=the heights are correlated.

If the second point is on a different block, they are not correlated.

The tendency will be that for small distances you will almost always remain on same block, whereas for large distances you will end up on a different block.

Hence the correlation function will have large values for small distances, and small values for large distances. The critical distance where you cross from one limit to the other will be the size of the block.

4. Feb 9, 2013

### Darwin123

I have seen a concept like what you describe which in the paper was called coherence distance. I can’t be sure if this is precisely the same concept that you describe. However, there is an article that is available on Research Gate.

The study was investigating multilayer structures. The substrate was assumed to have a vertical height that was randomly varying. The layers of the material were assumed share the same thickness, with positions that followed to randomly varying height. This type of structure was called “correlated roughness”, as the position of each interface was correlated with the one below it.

The standard deviation characterizes the width of the distribution. How the width statistically characterizes the spread depends on the functional form of the distribution.

The Gaussian distribution is not necessary in this description. In fact, the experimental data indicated that the random distribution of heights was basically a uniform function. One could say that the random distribution was a square wave rather than a Bell curve. For purposes of this discussion, it doesn’t matter whether the heights are distributed normally or squarely.

The “correlated roughness” varies with horizontal scale. Every value of correlated roughness has a certain horizontal distance over which the roughness is effectively constant. Generally, the standard deviation of heights slowly increases with the horizontal distance over which one takes the mean and standard deviation. Hence, with every measurement of mean height there is an upper bound to the horizontal distance over which that mean and standard deviation are valid.

The article discussed measurements of xray reflectivity, from statistical moments of height were determined. The hypothetical upper limit over which the determined moments of the height was valid was designated the “coherence distance”.

The illuminated area has to be less than the coherence distance in order for the calculated standard deviation to be correct. I think this concept is somewhat analogous to what you are asking.

I would consider this article mainstream. This article was refereed. It applied to a specific set of data which you can look at. I don’t know precisely what you are asking for, but look it over. Maybe it could help.

“Multilayer Roughness Evaluated by X-ray Reflectivity” by D. L. Rosen, D. Brown, J. Gilfrich and P. Burkhalter. Journal of Applied Reflectivity 21, 136-144 (1988).

5. Feb 10, 2013

### M Quack

Diffuse scattering can also be used to evaluate roughness. See e.g.

S. K. Sinha et an, Phys. Rev. B 38, 2297–2311 (1988)
X-ray and neutron scattering from rough surfaces

http://prb.aps.org/abstract/PRB/v38/i4/p2297_1

6. Feb 10, 2013

### Darwin123

My mental physical picture of "horizontal correlation length" corresponds closer to surface curvature than to surface roughness. In both light scattering and xray reflectivity measurements of surfaces, the curvature of the surface has to be distinguished from the roughness. The autocorrelation function often increases with the curvature of the surface.

This may be the "simple picture" that you are asking for. Roughness exists on what one could refer to as a microscopic length scale. Usually it is randomly distributed although not always. Even when it is random, the normality of the distribution is not critical. Curvature exists on what one could refer to as a macroscopic length scale. Curvature is more likely to be "deterministically" distributed.

Think of a glass lens. The surface of the glass lens is characterized by a curvature and a roughness. The curvature determines the "lens properties" and the roughness determines the "scattering processes". The correlation function is more analogous to curvature than to roughness.

While doing research on vertically correlated roughness, I had to characterize the curvature of the surface in some way. I had a problem because the displacement of height due to curvature over large illuminated distances was greater than the displacements in height due to roughness. It turned out that there was an "autocorrelation length" that was closer related to macroscopic curvature than than random variation. Random variation is what some call roughness.

The article is:
“Multilayer Roughness Evaluated by X-ray Reflectivity” by D. L. Rosen, D. Brown, J. Gilfrich and P. Burkhalter. Journal of Applied Reflectivity 21, 136-144 (1988).

The parts that you may be interested in regard the characterization of curvature. There are other articles out on roughness, many of them better, but I have a bias toward this one.

Dear Moderators. The article passed through the referee process. It is cited in some later papers. No one has seriously refuted it. So although it is "my theory", one can also call it "mainstream science". I do know what I am talking about, even on those occasions that I am wrong.