Maximum acceleration of an alpha particle?

[Orthoceras]

TL;DR Summary

Maximum acceleration of an alpha particle that is backscattered by a heavy atom?

I would like to estimate the maximum acceleration (or deceleration) of an alpha particle that is backscattered by a heavy atom, like in Rutherford backscattering. I am interested in the order of magnitude, not in a precise value. I am assuming the collision is elastic.

The kinetic energy of the free alpha particle is E. The closest distance of the alpha particle to the atom's nucleus, [itex]r_0[/itex], is found by equating E to the electrostatic potential energy: [itex] E = \frac{f q Q}{r_0} \Rightarrow r_0 = \frac{f q Q}{E}[/itex]
The acceleration (or deceleration) at [itex]r = r_0[/itex], due to the repulsive Coulomb force, is: [itex]a_0 = \frac{F}{m} = \frac{f q Q}{{r_0}^2 m} = \frac{f q Q}{\left(\frac{f q Q}{E}\right)^2 m} = \frac{E^2}{f q Q m} [/itex]

For example, if the alpha particle was generated by the decay of Po-210, with E=5.4 MeV, and the scattering nucleus is heavy compared to the alpha particle (for example, another polonium atom) then [itex]r_0 = 4⋅10^{-14} m[/itex] and [itex]a_0 = 3⋅10^{27} \frac{m}{s^2}[/itex]

However, I suppose a quantum mechanical wave does not have an acceleration, so r should be at least 10 times the de Broglie wavelength (assuming the de Broglie wavelength of a particle indicates the length scale at which wave-like properties are important for that particle). I tried a few r values and found that [itex]r=10 \lambda_{Broglie}[/itex] at [itex]r=1.7 r_0[/itex]. Then the acceleration is [itex]a = 1⋅10^{27} \frac{m}{s^2}[/itex]. That is my estimate for the order of magnitude of the maximum acceleration of this 5.4 MeV alpha particle.

Is this a somewhat valid estimate?

 

[Orthoceras]

However, I suppose a quantum mechanical wave does not have an acceleration, so r should be at least 10 times the de Broglie wavelength (assuming the de Broglie wavelength of a particle indicates the length scale at which wave-like properties are important for that particle). I tried a few r values and found that [itex]r=10 \lambda_{Broglie}[/itex] at [itex]r=1.7 r_0[/itex]. Then the acceleration is [itex]a = 1⋅10^{27} \frac{m}{s^2}[/itex]. That is my estimate for the order of magnitude of the maximum acceleration of this 5.4 MeV alpha particle.

Is this a somewhat valid estimate?

An edit of that paragraph, to provide more clarity and correct a mistake:

However, the speed is zero at [itex]r=r_0[/itex], so the de Broglie wavelength is infinite, the alpha particle is a quantum mechanical wave, and I suppose a wave does not have an acceleration. Assuming the de Broglie wavelength of a particle indicates the length scale at which wave-like properties are important for that particle, r should be at least 10 times the de Broglie wavelength. The de Broglie wavelength is [itex]\lambda=\frac{h}{p}[/itex] and [itex]p=\sqrt{2 m (E-\frac{kqQ}{r})}[/itex]. Then [itex]r\approx 10\, \lambda[/itex] at [itex]r=2 r_0[/itex], and the acceleration is [itex]a = \left(\frac{r0}{r}\right)^2a_0 = 0.7⋅10^{27} \frac{m}{s^2}[/itex]. That is my estimate for the order of magnitude of the maximum acceleration of this 5.4 MeV alpha particle.

Is this a somewhat valid estimate?

 

[kuruman]

You are looking for an order of magnitude estimate. I don't understand why you are not content with the particle formulation of the answer. Why bother with the wave formulation in which you know that "maximum acceleration" is an ill-defined concept?

 

[Orthoceras]

I bother with the wave formulation because of the next question. I was actually curious about the highest acceleration of objects present in our environment. The average human body contains 0.1 femtogram op Po-210, with an activity of 20 Bq, so many 5.4 MeV alpha particles are produced all the time inside us. The calculation of the maximum acceleration is the same as the one I used for Rutherford backscattering.

In this new case the wave is the initial form of the alpha particle, so it can't be ignored. My question is, at which distance do they transform from wave to particle, so at which distance is the concept of acceleration applicable? Would that be when [itex]r\approx 10\, \lambda[/itex], or at its first recoil action on the nucleus, or at its first interaction with an electron of the atom?

Even though an exact "maximum acceleration" is an ill-defined concept, I would expect that it should somehow be possible to estimate the order of magnitude.

 

[Vanadium 50]

Why are you considering alphas and not betas?

 

[Orthoceras]

In alpha decay, the initial speed of the particle is known, it is zero. In beta decay, the initial speed is unknown. As a result I was able to calculate the initial acceleration of the particle in alpha decay, but not in beta decay..

Also, Rutherford has shown that an alpha particle is able to penetrate the atom as a projectile. An electron that penetrates an atom is less simple.