Quantized Vortices in Superfluid Rotation

[Astrofiend]

I am trying to derive an expression for the number of quantized vortices created in superfluid He-4 when a container of radius R is rotated at angular velocity 'omega'.

I have completed an earlier part of the question, whereby I showed that the velocity of the superfluid as a function of r was v(r) = (n*hbar)/(m*r), where n is an integer, m is the mass of a helium 4 atom and r is the radial distance from the vortex core, when the container was rotated at just enough angular velocity to produce a single vortex. I'm not sure if this is related, but thought I'd slip it in. At higher rotational rates, the vortex breaks up and forms a number of smaller vortices based upon the rotational velocity of the container - and I need to find an expression that predicts this amount (with a few simplifying assumptions).

I have no idea how to go about this though or what the expression may end up looking like - and I've been dwelling on it for a long time! Any helping hands, pointers in the right direction, hints or tips would be greatly appreciated. I am at a complete loss!

 

[NateD]

The expression you are looking for is called the "vortex density". It is defined as the number of vortices per unit area and can be calculated using the equation: Vortex Density = (2*m*R*ω)/(hbar*n) where m is the mass of a helium 4 atom, R is the radius of the container, ω is the angular velocity of the container, n is an integer, and hbar is the reduced Planck constant.