Cooling by Adiabatic Demagnetisation

Definition: The cooling produced by reducing the magnetic field acting on a thermally isolated paramagnetic substance is called cooling by adiabatic demagnetisation.

Principle

A suitable paramagnetic salt, such as gadolinium sulphate, is placed in a thermally insulated glass tube. The tube is connected to a helium chamber and a vacuum system.

The cooling assembly is surrounded by liquid helium. Additional cryogenic shielding may be used to minimize heat transfer from the surroundings.

The sample is placed between the poles of a strong electromagnet. A magnetic field is therefore applied to the paramagnetic salt.

The temperature can be determined from the magnetic susceptibility of the paramagnetic material.

According to Curie's law, the magnetic susceptibility of a paramagnetic substance is inversely proportional to its absolute temperature:

\[ \boxed{ \chi=\frac{C}{T} } \]

where \(C\) is the Curie constant.

Method

1. The paramagnetic salt is brought into thermal equilibrium with liquid helium.
2. A strong magnetic field is applied to the salt.
3. The magnetic moments become partially aligned.
4. The salt is allowed to reach thermal equilibrium while the magnetic field is maintained.
5. The sample is thermally isolated from the surroundings.
6. The magnetic field is then reduced under approximately adiabatic conditions.
7. The temperature of the paramagnetic salt decreases.
8. Extremely low temperatures can therefore be obtained.

Thermodynamics of Adiabatic Demagnetisation

For a paramagnetic substance obeying Curie's law,

\[ \chi=\frac{I}{H}=\frac{C}{T} \]

we obtain

\[ I=\frac{CB}{\mu_0T},   \because H\approx\mu_0/B, \]

Differentiating with respect to temperature at constant \(B\),

\[ \left( \frac{\partial I}{\partial T} \right)_B = -\frac{CB}{\mu_0T^2} \]

Hence,

\[ \left( \frac{\partial S}{\partial B} \right)_T = -\frac{CB}{\mu_0T^2} \]

Entropy Equation

Let \(C_B\) be the specific heat capacity per unit mass at constant magnetic field and let \(\rho\) be the density of the specimen.

The heat capacity per unit volume is therefore

\[ \rho C_B \]

The entropy differential can consequently be written as

\[ dS= \frac{\rho C_B}{T}\,dT - \frac{CB}{\mu_0T^2}\,dB \]

Adiabatic Condition

For adiabatic demagnetisation,

\[ dS=0 \]

Therefore,

\[ \frac{\rho C_B}{T}\,dT = \frac{CB}{\mu_0T^2}\,dB \]

Multiplying by \(T^2\),

\[ \rho C_B T\,dT = \frac{CB}{\mu_0}\,dB \]

Hence,

\[ T\,dT = \frac{C}{\mu_0\rho C_B}B\,dB \]

Let

\[ \boxed{ A= \frac{C}{\mu_0\rho C_B} } \]

Then,

\[ T\,dT=A B\,dB \]

Integration

Integrating between the initial state \((T_1,B_1)\) and final state \((T_2,B_2)\),

\[ \int_{T_1}^{T_2}T\,dT = A\int_{B_1}^{B_2}B\,dB \]

Therefore,

\[ \frac{T_2^2-T_1^2}{2} = \frac{A}{2} (B_2^2-B_1^2) \]

Hence,

\[ \boxed{ T_2^2-T_1^2 = A(B_2^2-B_1^2) } \]

or

\[ \boxed{ T_2^2 = T_1^2+ A(B_2^2-B_1^2) } \]

Complete Demagnetisation

If the final magnetic field is reduced to zero,

\[ B_2=0 \]

Therefore,

\[ T_2^2=T_1^2-A B_1^2 \]

Hence, the final temperature is

\[ \boxed{ T_2= \sqrt{T_1^2-A B_1^2} } \]

Physical interpretation: During adiabatic demagnetisation, the magnetic field is reduced while the sample is thermally isolated. The decrease in magnetic ordering is accompanied by a decrease in thermal energy, resulting in a fall in temperature.

\[ \boxed{ \text{Compression} \rightarrow \text{Cooling} \rightarrow \text{Expansion} \rightarrow \text{Regeneration} \rightarrow \text{Liquefaction} } \]