Any alpha coming in inside this circle of radius "b" around the atom will scatter out beyond angle theta.
This means that
(the # of alphas scattered beyond
)/sec
=
[the right side is the (# incident/sec/area * area)]
Now generalize this to a real foil:
(# alphas observed beyond
)/sec
=
* (total # of atoms struck)
=
*(# atoms/unit volume)*(Volume of foil being struck)
=
where A = area of beam, and t=thickness of foil, and M is the gram formula weight of the target material = the number of grams/mole.
Finally, the fraction of alphas scattered beyond theta is this last equation divided by the total number of alphas hitting the foil,
(where I used my earlier result to eliminate b)
Everything here is experimentally measurable. Geiger and Marsden counted 100,000 flashes, and confirmed this equation.
You can go one step further. If this is the fraction that scatter beyond
theta, you can also differentiate with respect to
to find the fraction scattered between
and
.
If you then divide by the total area of the screen struck by just these
particles, you find the
(fraction of alphas scattering into angle
) / unit area
This is just a bit of algebra/calculus/geometry (another "few lines of math", and you get Rutherfords famous scattering formula,
is the (# of alphas / unit area) reaching the screen at angle
.
Ni is the total number of incident alphas.
r is the distance from the foil (target) to the screen
A better picture than I can draw is found in the appendix to Ch. 4 of Beiser.
This formula is an incredibly concrete, detailed prediction, based only on the
simple idea that the atom has a small, heavy, charged nucleus. Geiger and
Marsden painstakingly verified the dependence on t,
,
KE, and angle. It worked great!
This formula demonstrates quantitatively that Rutherfords picture of the atom was very accurate. It also gives us a way to learn something about the size of the nucleus! Consider those alphas that bounce backwards, at 180 degrees. These are the ones that must have come closest of all to the nucleus. At the moment they turn around,
KE = (1/2) mv^2 = (2e)(Ze)/R,
where R is the distance of closest approach. From this formula,
R = (2 Z e^2)/(1/2 m v^2)
For Rutherford's experiment, the initial KE was 7.7 MeV, so
For gold, Z=79, so R=3*10^-14m. Since scattering is still behaving like the nucleus is a "point" (that's built into the analysis, which is working), it must at least be smaller than this R. This gives a limit on the size of the nucleus. (In reality, the radius of a gold nucleus is about 1/5 of this.)
As Z gets smaller, so does R, and for Aluminum, the nice predicted behavior broke down, because the alpha particle began to make it into the nucleus!
Here is the Next lecture
Back to the list of lectures