HandsOn 13  Growing a Pattern in the Laboratory
IV. Data Analysis
It is likely that
the aggregate you grow does not appear to be a solid disk (most likely,
they will look something like Figures 4.4  4.7).
We want to measure the fractal dimension D of this aggregate. To estimate its
dimension, we use the circle method, described in HandsOn 7. For the radius
r of different circles, we substitute the approximate radius of the aggregate
at different times during its growth. Instead of counting boxes inside a circle,
we calculate the number N of copper atoms. If the deposit is a fractal, we expect
that the number of copper atoms N within a radius r to be

(4.1) 

(4.2) 
Before going further, discuss with your partners how you might measure the number of copper atoms that have been deposited at any time. Then read the following procedure:
(b) Divide this charge deposited by the charge q on each copper atom (3.2 x 10^{19} coulomb).
The result is the number N of ions
deposited in the time interval t:

On loglog paper we plot the quantity N versus the quantity r. If you don't
have loglog paper, you can use ordinary linear graph paper and press the
logarithm button on your calculator to make exactly this same plot. If
the resulting data appear to lie on a straight line, the growth is fractal
in nature, and the slope of the line D, is the fractal dimension, as
shown in Eqn. 2.
3. Using loglog graph paper or a calculator and linear graph paper or a computer
plotting program, plot logN versus logr. If the data appear to lie on a straight
line, measure the slope (the fractal dimension of the aggregate).
4. What is the total mass of the aggregate you grew?
5. In your experiment, the typical thickness of the aggregate is about 60
micrometers. The density of Cu metal is 8.92 gm/cm^{3}. What
would be the radius of a solid copper disk with the same mass as your final
deposit.