Probability and Alpha Radioactivity

Alpha radioactivity occurs when the alpha particle tunnels out of the nucleus. As we have previously seen the probability of tunneling is dependent on three parameters:
1) The energy of the object doing the tunneling (previously this was electrons, in this case it is the alpha particle)
2) The width of the barrier
3) The height of the barrier

If we imagine an alpha particle bouncing around a nucleus, it is a quantum mechanical process governed by probability. Each time it reaches vertical potential energy boundary there is a small probability tunneling.

Lets take a small diversion to look at probability in a more familiar example. If you took dice, say you had 15. You put them in a cup and then rolled them onto a table. Remove all of the dice with a number 6 on the top face. Put the remainder back into the cup and repeat the process until you have no dice remaining.

If you have this many dice at home - try it. Otherwise go to the kitchen and get a handful of corn kernels (popcorn - unpopped). Count the corn kernels and the put them into a cup and roll them onto a table. Remove all the corn kernels pointing north. Put the remainder back into the cup and repeat the process until you have no corn remaining.

Complete the table below:

Throw	Corn kernels remaining
1
2
3
4
5
6

Draw a rough graph of throws (x axis) vs. kernels remaining (y axis). You should see an exponential function, this is because the number remaining depends on number that you started with.

Now lets have a look at an example for alpha radioactivity.

Number of particles to start 1000
Time to traverse nucleus 1.1 seconds
Probability of tunneling 0.23

In this example the number of particles is the number of radioactive nuclei, the effects of particle energy, barrier width and barrier height are represented by the tunneling probability. The following table charts over time the radioactive decay of these 1000 nuclei.

Approach to Edge	Number Approaching Edge	Number Leaving Nucleus	Time
1	1000	230	1.1
2	770	177	2.2
3	593	136	3.3
4	457	105	4.4
5	352	81	5.5
6	271	62	6.6
7	208	48	7.7
8	160	37	8.8
9	124	28	9.9
10	95	22	11
11	73	17	12.1
12	56	13	13.2
13	43	10	14.3
14	33	8	15.4
15	26	6	16.5
16	20	5	17.6
17	15	4	18.7
18	12	3	19.8
19	15	4	20.9
20	7	2	22
21	5	1	23.1
22	4	1	24.2
23	3	1	25.3
24	2	1	26.4
25	2	0	27.5
26	1	0	28.6
27	1	0	29.7
28	1	0	30.8
29	1	0	31.9
30	1	0	33
31	0	0	34.1

Using either a piece of graph paper or an spreadsheet program make a graph of this data time (x axis) vs. number remaining (y axis). You should and exponential decrease in number of alpha particles coming out and in number of nuclei left behind.