Closest approach of alpha particle - two separate methods

• etotheipi
In summary, two methods are discussed for solving a problem with different answers. Method 1 assumes the larger nucleus does not move and equates energies before the collision and at the point of closest approach, while method 2 considers the movement of the larger nucleus and uses conservation of linear momentum and another energy calculation. It is recommended to use method 2 unless the movement of the nucleus is negligible, as the two methods may give different answers. A suggestion is given to determine the significance of the energy gained by the larger mass in order to determine which method to use.
etotheipi
Homework Statement
Find the distance of closest approach of an alpha particle and nucleus (of arbitrary atomic number Z)
Relevant Equations
##E_{k}##, relative velocities
I've found two methods for doing this problem and they give different answers.

Method 1: Assume the larger nucleus does not move, and simply equate energies before the collision and at the point of closest approach:$$\frac{1}{2} m v^{2} = \frac{qQ}{4\pi\epsilon_{0}r}$$

Method 2: Assume the larger nucleus is capable of moving; the distance of closest approach occurs when the relative velocity of the two particles is zero. From this, we can use CLM and then do another energy calculation:

Conservation of Linear Momentum:
$$m_{\alpha}v_{\alpha} = (m_{\alpha} + M_{nucleus})v_{common}$$Energy:
$$\frac{1}{2} m_{\alpha} v_{\alpha}^{2} = \frac{1}{2} (m_{\alpha} + M_{nucleus})v_{common}^{2} + \frac{qQ}{4\pi\epsilon_{0}r}$$It makes sense that the two methods give different answers for ##r##, however this is a very common exam question, and it is never usually stated whether we permit the larger nucleus to move or not. So, I was wondering which approach I should use for this and similar problems. Thank you!

Unless you can assume the nucleus is large enough that its movement is negligible you must use method 2. In the case where the nucleus is very large, methods 1 and 2 should give approximately the same answer.

etotheipi
PeroK said:
Unless you can assume the nucleus is large enough that its movement is negligible you must use method 2. In the case where the nucleus is very large, methods 1 and 2 should give approximately the same answer.

Thank you, this is essentially what I was thinking. It can be quite hard though to determine whether the assumption is justified or not. For instance, in the one below, I'd assume since silver is about 30x as massive as helium we could get away with the simpler method.

I suppose the only way to get a better feel for when certain assumptions are justified is just by doing lots of practice problems.

Here's an idea. In this collision, the energy gained by the large mass is approx ##\frac{m}{M}## of the energy of the system. Is that significant? In the above problem, this ratio is about ##4/50## = 8%.

So, do you need the answer to within 10%? If not, you can do a quick calculation by method 1.

etotheipi
PeroK said:
Here's an idea. In this collision, the energy gained by the large mass is approx ##\frac{m}{M}## of the energy of the system. Is that significant? In the above problem, this ratio is about ##4/50## = 8%.

So, do you need the answer to within 10%? If not, you can do a quick calculation by method 1.

That's a nice rule of thumb!

1. What is the closest approach of an alpha particle?

The closest approach of an alpha particle refers to the minimum distance between the alpha particle and the nucleus of an atom during a collision or interaction.

2. How is the closest approach of an alpha particle calculated?

There are two main methods for calculating the closest approach of an alpha particle: the classical approach and the semi-empirical approach. The classical approach uses the Coulomb's law to calculate the closest approach, while the semi-empirical approach takes into account the quantum mechanical effects.

3. What is the difference between the classical and semi-empirical methods?

The main difference between the classical and semi-empirical methods is the level of complexity. The classical approach is simpler and assumes that the particle and nucleus are point charges, while the semi-empirical approach takes into account the finite size and energy levels of the alpha particle and nucleus.

4. What factors impact the closest approach of an alpha particle?

The closest approach of an alpha particle is affected by several factors, including the initial kinetic energy of the alpha particle, the charge and size of the nucleus, and any intervening materials that may alter the trajectory of the alpha particle.

5. Why is the closest approach of an alpha particle important?

The closest approach of an alpha particle is important in understanding the behavior and interactions of particles at the atomic level. It can provide valuable insights into nuclear reactions and the structure of atoms, which have implications in fields such as energy production and medical imaging.

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