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15.3: Adiabatic Demagnetization

Last updated

Sep 10, 2020
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- 15.2: Adiabatic Decompression
- 15.4: Entropy and Temperature

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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

$\left(\frac{\partial S}{\partial T}\right)_{B}\left(\frac{\partial T}{\partial B}\right)_{S}\left(\frac{\partial B}{\partial S}\right)_{T}=-1. \label{15.3.1}$

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, $dQ = C_BdT$ . Here I am taking S to mean the entropy per unit volume, and C_B 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 $\left(\frac{\partial S}{\partial T}\right)_{B}=\frac{C_{B}}{T}$ .

The Maxwell relation corresponding to $\left(\frac{\partial S}{\partial P}\right)_{T}=-\left(\frac{\partial V}{\partial T}\right)_{P}$ is $\left(\frac{\partial S}{\partial B}\right)_{T}=\left(\frac{\partial M}{\partial T}\right)_{B}$ . Thus Equation $\ref{15.3.1}$ becomes

$\left(\frac{\partial T}{\partial B}\right)_{S}=-\frac{T}{C_{B}}\left(\frac{\partial M}{\partial T}\right)_{B}$ .

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 $\left(\frac{\partial M}{\partial T}\right)_{B}=-\frac{a B}{T^{2}}=-\frac{M}{T}$ . Equation 15.3.2 therefore becomes

$\left(\frac{\partial T}{\partial B}\right)_{s}=\frac{M}{C_{B}}.$

You should check the dimensions of this equation.

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

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