Simple relativistic harmonic oscillator

[photonist]

In the attached file, I have formulated a simple one dimensional harmonic oscillator and solved the model numerically. Such a model might represent a simple reaction coordinate along which a liquid drop actinide nucleus might split after absorbing a neutron. Clearly the complete model involves multiple modes and highly non-linear behaviour of the Hook's Law spring constant. But at least the approach this problem seems relatively simple.

 

[bcrowell]

Hi, photonist,

Welcome to PF!

Cool idea! However, I don't think the nuclear-physics application is right.

Fist off, the mass parameter is not going to be the mass defect. This kind of thing is generically known as an "inertial parameter" in nuclear physics, and it is very difficult to calculate. There are various approximations that can be written down for them, based on, e.g., low-amplitude oscillations of a classical, nonviscous liquid drop. But in reality, these approximations are only good to within about an order of magnitude. One reason is that nuclei are superfluid in the ground states. The only thing that really *is* straightforward to calculate is the inertial parameter after scission, which is just what you would expect classically for two separate bodies.

In the case of spontaneous fission, the order-of-magnitude theoretical uncertainty in the inertial parameter swamps any relativistic corrections. The relativistic corrections are quite small, since the fragments are not highly relativistic.

In addition, the anharmonicities are huge, especially when you're talking about fission (as opposed to a simple vibration of the nuclear shape about its equilibrium). Also it's a quantum-mechanical system, not a classical one. And nuclear vibrations at small amplitudes are typically not very highly collective, which means that it isn't really very accurate to think of them as pure shape oscillations; they're more like somewhat coherent superpositions of some small number of different single-particle excitations.

-Ben

 

[photonist]

HI Ben:
Thanks for your comments. Generally I agree with you--for one thing, the motion I posit is too simple.

However, the problem is interesting considering that fission in the odd actinides is occasioned by a thermal neutron--the energy gain is tremendous!

It is true that the droplet can be considered a superfluid and the Hook's Law constant is a measure of the separation of the neutron and proton fluid sub-components. But even so, when the thermal neutron is absorbed there has to be a path along a reaction coordinate (just as in chemical kinetics) established from the chaotic motion. A better approach would be to consider just how that is accomplished.

But the reason I looked into this problem was not to contribute to the liquid drop model, but rather to establish for myself a coherent special relativistic account of the harmonic oscillator problem to correct errors I found on the web. The liquid drop model appeared to be a possible case where the oscillator model would apply.

The novel feature of increasing odd harmonics as the maximum potential energy approaches the 'rest' energy suggests a decomposition mode. As the ratio approaches 1, the motion becomes 'unbound' (subject to the caveats mentioned).

Curious.

 

[photonist]

Ben:

You've pretty much disposed of fission as a potential candidate for my relativistic harmonic oscillator. There is of course plenty of MeV physics for which such analysis might be useful.

But what about a strongly optically dispersive medium on the short/long wavelength side of the absorption/gain line in which the refractive index can become really large. In effect the speed of light within a certain bandwidth will be greatly slowed. If that were the case, then harmonic oscillator radiators might exhibit 'relativistic' effects based on the local speed of light.

 

[bcrowell]

Ben:

You've pretty much disposed of fission as a potential candidate for my relativistic harmonic oscillator. There is of course plenty of MeV physics for which such analysis might be useful.

But what about a strongly optically dispersive medium on the short/long wavelength side of the absorption/gain line in which the refractive index can become really large. In effect the speed of light within a certain bandwidth will be greatly slowed. If that were the case, then harmonic oscillator radiators might exhibit 'relativistic' effects based on the local speed of light.

The c in relativity doesn't change based on the index of refraction of a medium.

You might be interested in this:

http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/SimpleElasticity.html