Growing a Pattern in the Laboratory

HandsOn 13 - Growing a Pattern in the Laboratory

IV. Data Analysis

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

N = cr^D,
(4.1)
where c remains constant as the fractal grows and D is the dimension of the pattern. If we take the logarithm of both sides of this equation we get

log N = log(cr^D) = D log r+log c.
(4.2)
To make use of this equation to find the dimension D, you need to determine the radius r of the pattern at specific times during the growth and the total number N of copper atoms in the pattern at these times. You measured the radius directly several times during the experiment. But what about number N of copper atoms?

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:

1. Use the following two steps to find the number $\Delta$ N of ions deposited in each time interval $\Delta$ t (where $\Delta$ means "change''):

(a) Find the charge deposited during that time interval. This is given by the expression I $\Delta$ t, which has the units (coulomb/second) times second = coulomb.

(b) Divide this charge deposited by the charge q on each copper atom (3.2 x 10^-19 coulomb). The result is the number $\Delta$ N of ions deposited in the time interval $\Delta$ t:

$\displaystyle \Delta$ N = $\displaystyle {I\Delta t\over q}$ .

You probably noticed that the current varies over each time interval.

Q4.9: Which is the "right'' value of the current to use for a given interval? The value at the beginning? The value at the end? Would it make sense to use the average?

2. Since you can compute $\Delta$ N for each time interval, you can do a running sum, adding the increase $\Delta$ t for the most recent time interval to the number summed from all previous time intervals. This gives the total number of ions deposited during any total time t. This is N(t), the total number of ions deposited as a function of time. But you can also measure the radius r at time t. It follows that you know N(r), the total number of ions deposited as a function of r.

On log-log paper we plot the quantity N versus the quantity r. If you don't have log-log 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 log-log 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/cm³. What would be the radius of a solid copper disk with the same mass as your final deposit.

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Next: 4.2 - More on Fractal Dimension