Roughness Concepts
RMS, Correlation Lengths, and the
Height-Height Correlation Function
By Max Bloomfield, June 2006
How rough are these surfaces?
© 2006,Max Bloomfield
This amount should be the same as…
We can establish
some “average
height”. Call it the
mean plane.
…this amount.
Same for this surface
© 2006,Max Bloomfield
What does “rough” mean?
What we know intuitively about roughness:
• If the surface was exactly on the mean plane, it
would be perfectly flat or “smooth”.
• How “rough” a surface is should be some measure
of how different the surface is from perfect
smoothness (from the mean plane).
Can we use the average difference of the
surface from the mean plane?
© 2006,Max Bloomfield
Distances below the mean plane are negative (yellow arrows)
and distances above are positive (purple arrows). And we chose
the mean plane’s position so these values all add to zero! (We
did that when we used the rule that the amount of surface above
and below the mean plane should be equal.)
© 2006,Max Bloomfield
RMS roughness
• We use the traditional method for getting rid of
signs that cancel: by using the square of the
distance.
– Find the mean of the squared distances (making them all
positive values) and then take the square root!
• This is called the root mean square or RMS
roughness.
• RMS roughness is often given the symbol w.
© 2006,Max Bloomfield
RMS roughness turns out
to be good for comparing
how rough things are
when they have similar
structures.
For example, here B has
3 times the w of A, and C
has 6 times the w of A.
(That’s good, because B
and C are just A
stretched vertically by a
factor of 3 and 6
respectively.)
© 2006,Max Bloomfield
A
B
C
Which of these is rougher?
A
B
Just using RMS can be deceiving. Above, A and B have the same
RMS roughness! B was constructed by scaling A horizontally, so
they deviate from the mean plane by the same amount on average.
So why does B seem so much rougher? It’s the quicker variation in
the vertical as a function of horizontal position.
© 2006,Max Bloomfield
Horizontal Roughness
• We can guess that horizontal roughness must have to do
with how wide the bumps are. Are they thin and sharp
peaks, or wide and rolling hills?
• How do you measure how wide bumps are? How do you
even decide where a bump starts and ends?
• Reformulate that last question to “How far do I have to go
horizontally (on average) so that I’m no longer on the same
bump?”
© 2006,Max Bloomfield
Horizontal Correlation Length (ξ)
1. Pick a point on the surface. Call that horizontal position
a and call its height z(a).
z(a)
a
© 2006,Max Bloomfield
Horizontal Correlation Length (ξ)
1. Pick a point on the surface. Call that horizontal position
a and call its height z(a).
2. Now pick another point. Call its horizontal position b
and its height z(b). Call the horizontal distance between
them r.
b
z(a)
z(b)
a
© 2006,Max Bloomfield
Horizontal Correlation Length (ξ)
3. Average the squared difference in height for a large
number of a’s always choosing b so that it is a
distance r from the current a. Call that number H(r).
(I’ve shown 3 samples of a below, moving r in both
directions from each to get b’s. On a real surface,
you’d be able to look more than just 2 directions.)
b2
r b
2
r
© 2006,Max Bloomfield
a2
r
b1
a1 r
b1
b3
r
r
a3
b3
Three possible situations:
• If r is very small (a is so close to b that they are definitely on the
same “bump”) then z(a) and z(b) will tend to be close, so their
average square distance is small and based on r through the local
shape. So H(r) is small and depends on r and the graph of H(r)
increases with small r.
• If r is large (a is definitely on a different bump than b) then z(a)
and z(b) will be different by an amount only based on how much
the surface varies vertically on average (think w) and not on r.
Turns out H(r) = 2w2 for large r, so H(r) is flat for large r.
• If r is about the size of a bump, b will just start to lose its
relationship to (or “correlation with”) r, so a graph of H(r) should
bend down to be constant when r hits the characteristic size of
bumps. This is the “correlation length” or ξ (The Greek letter “xi”).
z(a)
z(b)
b
© 2006,Max Bloomfield
a
S tr
ong
d ep
en d
en c
eo
nr
2w2
ξ
© 2006,Max Bloomfield
Some things to remember:
• w can fool you. Very different structures can have the
same RMS roughness. Think sine wave versus sawtooth.
• ξ can fool you. The graph of H that I showed was for a
“vaguely sane surface”. You can imagine surfaces with
long range order built into them for which coming up with
a ξ would be almost meaningless.
• ξ isn’t exactly the “bump width”. By the very nature of our
noisy surface, “bump width” is not a well defined term.
• ξ is a function of what you know. It can change when you
measure more or less closely. Large scale scans can miss
higher frequency noise.
© 2006,Max Bloomfield