User:Kurzon/Scattering

Scattering

Rutherford's mathematical model

Considering the results of the above experiments, Rutherford published a landmark paper in 1911 where he proposed the existence of a strong electrical charge concentrated in a very small volume at the center of the atom. Rutherford developed a mathematical equation that modeled how the foil should scatter the alpha particles according (see the Rutherford scattering article for a detailed explanation).

$a={\frac {Ant\csc ^{4}{\tfrac {\phi }{2}}}{16r^{2}}}({\frac {2Q_{n}Q_{a}}{mv^{2}}})^{2}$

a = the number of alpha particles falling on unit area which are deflected through an angle Φ is given by
r = distance from point of incidence of α rays on scattering material
A = total number of particles falling on the scattering material
n = number of atoms in a unit volume of the material
t = thickness of the foil
Q_n = positive charge of the atomic nucleus
Q_a = positive charge of the alpha particles
m = mass of an alpha particle
v = velocity of the alpha particles

The 1913 experiment

In a 1913 paper, Geiger and Marsden describe a series of experiments by which they sought to experimentally verify the above equation that Rutherford developed. Rutherford's equation predicted that the number of alpha particles per minute a that will be observed at a given angle Φ should be proportional to:

1. csc⁴Φ/2
2. thickness of foil t
3. magnitude of central charge Q_n
4. 1/(mv²)²

Their 1913 paper describes four experiments by which they proved each of these four points.

This apparatus was described in a 1913 paper by Geiger and Marsden. It was designed to accurately measure the scattering pattern of the alpha particles produced by the metal foil. The microscope and screen were affixed to a rotating cylinder and could be moved a full circle around the foil so that they could count scintillations from every angle.

To test the first point—that a is proportional to csc⁴Φ/2—Geiger and Marsden built an apparatus that consisted of a hollow metal cylinder mounted on a turntable. Inside the cylinder was the metal foil and the radiation source (containing radon), mounted on a detached column which allowed the cylinder to rotate independently. A microscope with its objective lens covered by a fluorescent zinc sulfide screen penetrated the wall of the cylinder and pointed at the metal foil. By turning the table, the microscope could be moved a full circle around the foil, allowing Geiger to observe and count alpha particles deflected by up to 150°. Correcting for experimental error, Geiger and Marsden found that the number of alpha particles that are deflected by a given angle Φ is indeed proportional to csc⁴Φ/2.

This apparatus was used to measure how the alpha particle scattering pattern varied in relation to the thickness of the foil. The rotating disc in the center had six holes covered with foils of varying thickness.

Geiger and Marsden then tested how the scattering was affected by the thickness of the foil. They constructed a disc with six holes drilled in it. The holes were covered with metal foil of varying thickness, or none for the sake of control. This disc was then sealed in a brass ring between two glass plates. The disc could be rotated by means of a rod to bring each window in front of the alpha particle source. Geiger and Marsden experimented gold, tin, silver, copper, and aluminum. They found that the number of scintillations that appeared on the zinc sulfide screen was proportional to the thickness as long as said thickness was small. At larger thicknesses, the change in the velocity of the alpha particles skews the relation significantly.

Geiger and Marsden reused this apparatus to test the third point, that the scattering is proportional to the amount of positive charge in the nucleus (Q_n). Geiger and Marsden didn't know what the positive charge of the nucleus of their metals were (they had only just discovered the nucleus existed at all), but they assumed it was proportional to the atomic weight.

Papers

• Rutherford, Ernest (1911). "The Scattering of α and β Particles by Matter and the Structure of the Atom". Philosophical Magazine. Series 6. 21: 669–688. doi:10.1080/14786440508637080.

• Geiger, Hans; Marsden, Ernest (1913). "The Laws of Deflexion of a Particles through Large Angles". Philosophical Magazine. Series 6. 25 (148): 604–623. doi:10.1080/14786440408634197.