Modeling an Einstein solid that is coupled to a paramagnet

[Ron Burgundypants]

I'm working on a project at university to calculate the magnetocaloric effect of dysprosium. This will be done using a new technique designed at the university of which its not necessary to go into detail about. In short, the Dy is placed in a solenoid, through which a current runs, the current causes dipole alignment and a lowering of the magnetic entropy of Dy. Now assume we have Isentropic conditions, for the total entropy to stay constant the temperature entropy must increase to counter the magnetic entropy decrease, this is the basic idea of the MCE. More can be read below

https://en.wikipedia.org/wiki/Magnetic_refrigeration

The problem

I want to model the system so I can run some simulations, make some calculations and measure them afterwards in the real setup.

I think the system (Just the Dysprosium) can be modeled as an einstein solid and a paramagnet at the same time but I think its probably not that simple. I spoke to my supervisor about it and he agreed on the model but that there would also be some summing of the entropies of the different states of the system, so maybe there are some states that can be neglected to make the problem simpler, but I feel there is a lot more to this problem that I haven't thought of.

I first looked at the product of the multiplicities of an Einstein solid in the limit were q>>N (much more energy than the no. of oscillators) and a paramagnet. Is this a fair assumption to make? I get a very ugly expression which you can see by just taking the product of the multiplicity of the Einstein solid and the paramagnet, I won't bother posting it just yet.

I also have 10g of dysprosium, some quick calculations show that the amount of energy therein is greater than the number of particles by 10 orders of magnitude. Using Schroeders book ' Introduction to thermal physics' as a guide I see he discerns between 'large' and 'very large' numbers but its not entirely clear where the boundaries are. I know 10 orders is a HUGE amount of difference but is it reasonable to change the equations because of the difference between these numbers? Is it big enough?!?

So I know I'm not really asking a specific question but I would like to hear some thoughts, ideas, suggestions on the matter. How can I tweak the model, is it accurate, what other considerations have I missed?

Thanks guys

 

[Ron Burgundypants]

The effect of the phonons in the sample is something we haven't thought about, there would be some coupling but again, its a bit beyond my level.