Scattering formulas, what's the point?

[Erland]

I am currently redaing Goldstein's Classical Mechanics (3rd ed. 2002, with Poole and Safko). The sections 3.10-11 deal with scattering, of the type originally studied by Rutherford, and which led him to formulate his "planetary system" atom model.

Rutherford fired a beam of positively charged alpha particles against a thin gold foil, and could explain the results of the scattering of the particles with the model that the atom consists of a (relatively) heavy positively charged nucleus with light negatively charged electrons orbiting it at a (relatively) large distance. The deflections take place if and when the particles come close to the nucleus. The more general problem is to calculate the deflection pattern if a beam of particles is fired against an originally fixed center of force.
Goldstein makes great pains to calculate the so called cross section for scattering in a given direction, first assuming that the target (in Rutherford's case the nucleus) remains fixed all the time, and then when the target moves in reaction to the collision.

I don't see the point with these detalied calculations. It seems to me that the formulas will not apply in a real experiment such as Rutherford's:

1. In the calculations, it is assumed that the center of force is originally fixed. But, unless it is held fixed in a lattice or something like that, this can only be true before the interaction with the first particle in the beam which hits it. When the second, third etc. particle in the beam hits the center of force, it will move in reaction to prior collisions, in a chaotic manner making it unmanageable to calculate the scattering angles, and hence the cross section for scattering to be measured.

2. In Rutherford's experiment, why don't the particles in the beam interact with the electrons, and become deflected by such interactions?

3. Why don't the particles in the beam interact with each other, and become deflected by such repulsions?

 

[dauto]

1. The target atoms are fixed to a lattice and there are many more target atoms than incoming particle so the chances of two incoming particle hitting the same atom is negligible

2. The electrons are a bunch of light weights that scatter out of the way of the incoming particles with negligible effect on them

3.The beam density is too low so the particles in the beam are too far away from each other and their mutual interactions is also negligible.

 

[Erland]

Ok, thanks Dauto. One more thing, though:

2. The electrons are a bunch of light weights that scatter out of the way of the incoming particles with negligible effect on them

But the mass of an electron does not matter for the electrostatic force it exerts upon a particle, only its charge matters for this, and this charge is the same as the charge of a much heavier proton (although with opposite sign). On the other hand, the nucleus of a gold atom (which Rutherford used) contains 79 protons, so its charge is 79 times the charge of an electron and exerts a much greater force upon the incoming particle than any electron...

 

[dauto]

Ok, thanks Dauto. One more thing, though:


But the mass of an electron does not matter for the electrostatic force it exerts upon a particle, only its charge matters for this, and this charge is the same as the charge of a much heavier proton (although with opposite sign). On the other hand, the nucleus of a gold atom (which Rutherford used) contains 79 protons, so its charge is 79 times the charge of an electron and exerts a much greater force upon the incoming particle than any electron...

The electrons scatter OUT OF THE WAY, so they are never close enough to the incoming particles to have a measurable effect on them

 

[sardar]

I can understand your confusion and questions about the purpose of scattering formulas. However, these calculations serve a crucial role in understanding the behavior of particles and their interactions in various systems. Let me address your concerns one by one:

1. The assumption of a fixed center of force is a simplification made in order to solve the equations of motion and calculate the scattering patterns. While it may not perfectly reflect the real experimental conditions, it allows for a mathematical understanding of the underlying physics. In reality, the center of force may indeed move in a chaotic manner due to prior collisions, but the overall behavior of the particles can still be described using these formulas. Furthermore, in some cases, the center of force may remain fixed, such as in the case of a heavy nucleus in Rutherford's experiment.

2. The particles in the beam may indeed interact with the electrons in the target, but the scattering formulas are specifically designed to study the interactions between the beam particles and the center of force (in this case, the nucleus). The effects of interactions with electrons can be taken into account in more complex calculations, but the basic scattering formulas provide a starting point for understanding the behavior of the particles.

3. Similarly, the particles in the beam may interact with each other, but the scattering formulas focus on the interactions between the beam particles and the center of force. This is because the behavior of the particles in a beam is largely determined by the interactions with the target, and not necessarily with each other.

In summary, while the scattering formulas may not perfectly reflect the real experimental conditions, they serve as a useful tool for understanding the fundamental interactions between particles. They allow us to make predictions and calculations that can then be compared to experimental results and further refined. So, while they may seem abstract and detached from reality, they play a crucial role in our understanding of the physical world.