Rutherford scattering
The Rutherford scattering describes the scattering of charged particles on a charged scattering center. In the starting test, the dispersion of alpha- particles was evaluated in gold nuclei. The resulting particle trajectories are hyperbolas. The distribution of the scattered particles can infer the structure of the scattering center. This led to the conclusion that the positive charge in the atoms concentrated to a small space in the atom center. Until then, the model of J. J. was Thomson, in which the positive charge of the atom is homogeneously distributed in a sphere (Thomson cal model of the atom ). Participation in these experiments were Hans Geiger, Ernest Marsden and Ernest Rutherford. When considering the measurement results indicate that the mass of the atom is concentrated in a small nucleus, Rutherford reported to have said: "This is so unlikely as if you shoot with a gun on a cotton ball, and the ball bounces back. "
- 2.1 plausibility consideration of dependencies
Rutherford's scattering experiment (Manchester, 1909-1913 )
Design and Experimental Procedure
Into a lead block with an opening to one side, a radioactive substance is placed, which emits radiation: alpha-, beta - and gamma - radiation. Emerging from the opening in the lead block beams are guided by an electric field to separate them from each other. As a result, the negative electrons ( beta radiation ) to the positive pole and the positive helium nuclei ( alpha rays ) to be deflected to the negative pole, while the direction of uncharged photons ( gamma rays ) remains unchanged. The alpha radiation is perpendicular to a 0.5 micron thin gold film ( about 1000 atoms in a row) directed. The escaping radiation from the foil can be made visible afterwards with a fluorescent screen or an attached film. (Gold was used because it is already at that time could be processed with simple mechanical means into very thin layers and a high atomic mass has. )
Observation
- Almost all the alpha particles pass through unhindered the gold foil.
- Only about 1 of 100,000 alpha particles, the direction is changed.
- Larger scattering angles before coming here less and less, the greater the angle.
- Also scattering angle of about 90 ° there is, but extremely rare.
- Some alpha particles are backscattered.
For the observed distribution Rutherford developed the scattering formula described below.
Interpretation
The extremely rare deflection of the alpha particles and their angular distribution can be understood that only a very small center of mass is in the atoms, which is positively charged thereby. We call this the center of mass of the atomic nucleus. Since most of the particles pass through the gold foil without hindrance, there must be a large space between the cores. This result led to the Atomic Energy rutherford model. ( The electrons, which move in relation to the core diameter large empty space (vacuum) around the core, the core shield the positive charge from concentrated, so that the atomic outwardly appear neutral. )
Rutherford scattering formula
The Rutherford scattering formula are the so-called differential scattering cross-section (also referred to as cross-section ) in dependence on the scattering angle in the priority system:
The same formula in nuclear physically meaningful units:
So that the likelihood is described as being scattered particles impinge after deflection by the angle in the chamber angle.
In the formula further following variables are used:
In the pre-factor is arrived at by using the following variables:
Rutherford directed the Rutherford scattering formula from classical physics ago. A full quantum- mechanical treatment of the problem using the Born approximation shows that the Rutherford scattering formula in the first order is correct and quantum mechanical effects represent only small corrections. Another problem of the Rutherford formula is the limiting case for the differential cross section is infinite. However, small angles correspond to a large impact parameter. For very large impact parameters, however, the atomic electrons shield the nucleus. The only way to have a very small angle at small impact parameters is to increase the energy of the α particles. For very high energies, the charge distribution of the nucleus can no longer be assumed to be point-like. Then the form factor of the charge distribution is in addition to the data scattering formula. In addition, one can no longer assume that the scattering is only through electromagnetic interactions at high projectile energies. Approaching the two cores to a contact radius, the strong interaction plays an important role.
Plausibility consideration of dependencies
After Feynman rules is obtained for the scattering of a particle of the charge to a second particle of charge for the probability amplitude
Where the propagator has been neglected. According to Fermi's golden rule
Thus follows that
Derivation of the Rutherford scattering formula
Due to the action of the Coulomb repulsive
Results for the path of the alpha particle () is a hyperbola.
The semi-major axis a of the hyperbola can be derived from the approach
Determine, with the minimum distance of the alpha particle, when it collides with the central core. is dependent on the kinetic energy and can also be used for joints, which are not centrally applied. The impact parameter is the minimum distance of the alpha particle to the core, if it would fly on a straight on. In fact, the alpha particle is scattered by the angle. From the geometry of the hyperbola we obtain the following equations:
And thus
By deriving the last formula is obtained a relationship between the width of a hollow cone and the corresponding width of the deflection angle.
Is the particle density (atoms per volume) of the scattering material and the thickness of the film, as is the average cross-sectional area per atom experienced by the alpha particles when passing through the film. is also called the cross -section.
The probability in the ring of the hollow cylinder to land is then given by
From particles are scattered in the hollow cone. , The probability
Is the number of particles in the solid angle.
It follows:
The result is the probability
This is the Rutherford scattering formula. It indicates how high the probability of a particle is to be scattered into the solid angle.
Often the stray formula is given using the differential cross -section. It is a measure for the same probability.
It is
And thus
.
Comments
1) is not defined since there is a minimal deflection. This is assumed when the alpha particles thus moved apart at the edge of the circular cross -sectional area of the atom. For a larger impact parameter, the alpha particle is in the stray field of the neighbor atom and the deflection angle increases again. The following applies:
2) the integral over the probability distribution results in 1
3) is similar for the surface integrals