# Modeling an Einstein solid that is coupled to a paramagnet

• Ron Burgundypants
In summary, the student is working on a project to calculate the magnetocaloric effect of dysprosium. They first looked at the product of the multiplicities of an Einstein solid and a paramagnet, but they got an ugly expression. They also know that the amount of energy in dysprosium is greater than the number of particles by 10 orders of magnitude, and they are wondering if it is reasonable to change the equations because of the difference between these numbers.
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

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.

## 1. What is an Einstein solid?

An Einstein solid is a theoretical model used in statistical mechanics to understand the behavior of a system of particles with fixed energy levels. It assumes that the particles can only occupy discrete energy levels and cannot exchange energy with each other.

## 2. How is an Einstein solid coupled to a paramagnet?

In this model, the Einstein solid is coupled to a paramagnet by considering the energy levels of the solid as the magnetic moments of the paramagnet. This allows for the exchange of energy between the solid and the paramagnet, resulting in a more realistic model.

## 3. What is the significance of studying this model?

Modeling an Einstein solid coupled to a paramagnet allows us to understand the behavior of complex systems in which energy can be exchanged between different components. This model has applications in fields such as materials science, condensed matter physics, and thermodynamics.

## 4. How is the coupling between the solid and paramagnet represented mathematically?

The coupling between the solid and paramagnet is represented by a coupling constant, which determines the strength of the interaction between the two components. This constant is typically denoted as g in the equations describing the model.

## 5. What are some limitations of this model?

This model assumes that the coupling between the solid and paramagnet is weak and that the energy levels of the solid are evenly spaced. This may not accurately represent all systems, and more complex models may be needed to fully understand certain phenomena.

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