52 4 Mounded Surfaces
This autocorrelation function leads to an expression for the PSD that does not
diverge as α → 0, whereas the exponential model breaks down in this limit.
Also, the K -correlation model is able to give a rational expression for the
PSD if α = 1, whereas the exponential model gives a transcendental function
that is more difficult to analyze. It should be noted that in 1+1 dimensions,
the form of these autocorrelation functions is the same as in 2+1 dimensions
except for the substitution of a cos function for the Bessel function J
0
,which
is required to obtain an analytic expression for the PSD in 1+1 dimensions
as discussed in Sect. A.4.5.
In borrowing the self-affine height–height correlation function, the rough-
ness exponent α carries over into the mounded height–height correlation func-
tion, which may seem contradictory. The roughness exponent α was defined
in terms of the scaling behavior of a self-affine surface, and mounded surfaces
are not self-affine, thus it would seem as if α would not have any meaning
for mounded surfaces. However, recall that α reflects the short-range, or local
roughness of a surface. On length scales much smaller than the wavelength
λ, a mounded surface “appears” self-affine becuase there is no characteris-
tic length scale smaller than the wavelength, and the local roughness is well
defined in terms of the locally self-affine behavior of the mounded surface.
In fact, from the form of the height–height correlation function, if r λ,
J
0
(2πr/λ) ≈ 1, and the self-affine height–height correlation function is recov-
ered. Only when describing the long-range behavior (r ≥ λ) of a mounded
surface does the oscillatory term in the height–height correlation function be-
come significant. Nevertheless, the local surface is not truly self-affine because
scaling this surface beyond the wavelength destroys the self-affine scaling na-
ture of the morphology.
An expression for the local slope of a mounded surface can be extracted
from these model correlation functions using (3.21). The exponential model
gives, for α =1,
m =
w
√
2
ξ
1+
πξ
λ
2
. (4.4)
However, for α<1, the mounded exponential model gives the same local
slope as the self-affine exponential model. This occurs because the local slope
depends only on the small r behavior of the autocorrelation function. The
small r behavior of the exponential model for the autocorrelation function is
R(r) ≈ 1 −
r
2α
ξ
2α
−
π
2
r
2
λ
2
+ O
r
2+2α
. (4.5)
When α<1, only the first two terms in this expansion are significant when
evaluating the local slope because the term in r
2
is of too high an order.
However, when α =1,bothtermsinr are of the same order, and both
contribute to the value of the local slope. A similar result is obtained with the
K -correlation model for the autocorrelation function, as a Taylor expansion
will show similar small r behavior to (4.5). Therefore, according to this model,