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15.3: Adiabatic Demagnetization

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We are now going to do the same argument for adiabatic demagnetization.

We are going to calculate an expression for (T/B)S. The expression will be positive, since T and B increase together. We shall consider the entropy as a function of temperature and magnetic field, and, with the variables

Screen Shot 2019-07-08 at 12.11.09 PM.png

we shall start with the cyclic relation

(ST)B(TB)S(BS)T=1.

The middle term is the one we want. Let’s find expressions for the first and third partial derivatives in terms of things that we can measure.

In a reversible process dS=dQ/T, and, in a constant magnetic field, dQ=CBdT. Here I am taking S to mean the entropy per unit volume, and CB is the heat capacity per unit volume (i.e. the heat required to raise the temperature of unit volume by one degree) in a constant magnetic field.

Thus we have (ST)B=CBT.

The Maxwell relation corresponding to (SP)T=(VT)P is (SB)T=(MT)B. Thus Equation ??? becomes

(TB)S=TCB(MT)B

.

Now for a paramagnetic material, the magnetization, for a given field, is proportional to B and it falls off inversely as the temperature (that’s the equation of state). That is, M = aB/T. and therefore (MT)B=aBT2=MT. Equation 15.3.2 therefore becomes

(TB)s=MCB.

You should check the dimensions of this equation.

The cooling effect is particularly effective at low temperatures, when CB is small.


This page titled 15.3: Adiabatic Demagnetization is shared under a CC BY-NC license and was authored, remixed, and/or curated by Jeremy Tatum.

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