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Thermal Physics
Production of Low Temperature by Adiabatic Demagnetization
Adiabatic demagnetization is a thermodynamic technique used to attain temperatures close to absolute zero by rapidly reducing the magnetic field applied to a paramagnetic salt. The process relies on the entropy change associated with the alignment and randomization of atomic magnetic dipoles.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Basic qualitative understanding of paramagnetism, entropy, isothermal magnetization followed by adiabatic demagnetization, and cooling effects.
PGT extension
Rigorous thermodynamic treatment using Maxwell's relations, calculation of entropy change, Curie's law, and derivation of the final temperature expression.
Prerequisites
Concept of Temperature and Entropy
Law of Equipartition of Energy
Basic thermodynamics of adiabatic processes
Principle of Adiabatic Demagnetization
First proposed by P. Debye (1926) and W. Giauque (1927) to achieve temperatures below 1 K.
Relies on paramagnetic salts containing ions with unpaired electrons (e.g., gadolinium sulfate).
Total entropy of the salt is the sum of lattice entropy and magnetic entropy.
Two-Stage Cooling Cycle
Stage 1 (Isothermal Magnetization): The paramagnetic salt at a low initial temperature (e.g., using liquid helium bath at 1 K) is subjected to a strong magnetic field. Heat of magnetization is rejected to the surroundings.
Stage 2 (Adiabatic Demagnetization): The salt is thermally isolated from the bath, and the magnetic field is rapidly reduced to zero, causing the magnetic dipoles to randomize at the expense of thermal energy, lowering the temperature.
The lattice vibrations decrease as thermal energy is transferred to reorganize magnetic spins, leading to a net drop in temperature.
Role of Magnetic Entropy
In a strong magnetic field, magnetic dipoles align, lowering the magnetic entropy.
When the field is removed adiabatically, the total entropy remains constant, requiring the magnetic entropy to increase by absorbing energy from the thermal lattice reservoir, thus cooling the material.
At extremely low temperatures, lattice entropy becomes negligible compared to magnetic entropy.
Limitations in Reaching Absolute Zero
Absolute zero (0 K) cannot be reached in a finite number of steps in accordance with the Third Law of Thermodynamics.
Residual interactions between magnetic ions (internal molecular fields) prevent complete disorder even at zero external field.
Heat leaks from the experimental apparatus impose a lower bound on the minimum attainable temperature.
Derivations
Thermodynamic Relation for Adiabatic Demagnetization
Conditions: Reversible adiabatic process for a paramagnetic substance
Write the fundamental thermodynamic equation for a magnetic system: d$U = TdS + Hdm$, where H is the magnetic field intensity and m is the total magnetic moment.
Define the magnetic enthalpy or Gibbs-like potential: d$G = -SdT - mdH (using internal energy minus magnetization term)$.
Apply the Maxwell relation derived from exact differentials: (\frac{\partial S}{\partial H})_$T = (\frac{\partial m}{\partial T})_H$.
Use Curie's Law for paramagnetism: m = \frac{C H}{T}, where C is the Curie constant. Differentiate with respect to T: (\frac{\partial m}{\partial T})_$H = -\frac{C H}{T^2}$.
Substitute into the entropy change equation for an infinitesimal reversible process: d$S = (\frac{\partial S}{\partial T})_H dT + (\frac{\partial S}{\partial H})_T dH = \frac{C_H}{T} dT - \frac{C H}{T^2} dH$.
For an adiabatic process (d$S = 0): \frac{C_H}{T} dT = \frac{C H}{T^2} dH$, which integrates to give the temperature change.
Result: d$T = \frac{C H}{C_H T} dH$
Change in Temperature during Demagnetization
Conditions: From initial field H_i to final field $H_f = 0$, assuming constant heat capacity C_H
Start from the adiabatic condition: C_H \frac{dT}{T} = \frac{C H}{T} dH (multiplying both sides by T).
Integrate from initial state (T_i, H_i) to final state (T_f, $H_f = 0): \int_{T_i}^{T_f} \frac{dT}{T} = \int_{H_i}^{0} \frac{C}{C_H} \frac{H}{T^2} dT$... wait, using C_H dT = \frac{CH}{T} dH.
More precisely, integrate C_H d$T = m dH with m = C H / T: \int_{T_i}^{T_f} C_H dT = \int_{H_i}^{0} \frac{CH}{T} dH$... integration requires keeping T inside or using entropy constancy.
Using entropy constancy: S(T_i, $H_i) = S(T_f$, 0). Since magnetic entropy depends on H/T, the ratio of fields corresponds to the ratio of temperatures under specific approximations.
Result: $T_f = T_i \left(\frac{H_f}{H_i}\right) \text{ (approximate linear regime)}$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Curie's Law of Paramagnetism
$$M = \frac{C \cdot B}{T}$$
M = Magnetization, C = Curie constant, B = Magnetic induction, T = Absolute temperature M in A/m, B in Tesla, T in Kelvin
High temperatures and weak magnetic fields
First Law for Magnetic Systems
$$dU = T dS + H dm$$
U = Internal energy, T = Temperature, S = Entropy, H = Magnetic field, m = Magnetic moment Joules (J)
T_f, $T_i = Final$ and initial temperatures, H_f, $H_i = Final$ and initial magnetic fields Kelvin (K)
Ideal paramagnetic substance obeying Curie's law
Magnetic Heat Capacity
$$C_m = \frac{A}{T^2}$$
$C_m = Magnetic heat capacity$, A = Material constant J/K
Schottky anomaly / low temperature limit (T << 1 K)
Third Law Limit
$$\lim_{T \to 0} S = 0$$
S = Entropy, T = Temperature J/K
Pure crystalline substance at absolute zero
Work Done in Magnetization
$$W = - \int m \cdot dH$$
W = Work done on the magnetic system, m = Magnetic moment, H = Field Joules (J)
Isothermal processes
Graphs and experiments
T-S (Temperature-Entropy) Diagram of Adiabatic Demagnetization
Draw: A cycle consisting of an isothermal magnetization line (vertical drop in entropy at constant T_i) followed by an adiabatic demagnetization line (vertical drop in temperature at constant entropy S).
Interpret: Illustrates how isothermal compression removes heat at higher T, and subsequent adiabatic expansion along S = const drops the temperature to T_f.
Exam use: Frequently asked in PGT to evaluate understanding of thermodynamic cycles and entropy constancy.
Magnetization vs H/T (Brillouin Function)
Draw: A non-linear saturation curve plotting magnetization against H/T, showing linear behavior at small arguments (Curie region) and saturation at high fields/low temperatures.
Interpret: Demonstrates the deviation from simple Curie law at extremely low temperatures or high fields.
Exam use: Tested conceptually in TGT/PGT regarding paramagnetism limits.
Giauque-MacDougall Experiment Setup
Draw: Schematic of a paramagnetic salt pill suspended in a helium cryostat surrounded by an electromagnet and a low-pressure helium exchange gas chamber.
Interpret: Shows the practical isolation required to prevent heat leaks during adiabatic demagnetization.
Exam use: Used for descriptive questions on experimental realization of sub-Kelvin temperatures.
Solved numericals
TGT core
A paramagnetic salt at an initial temperature of 1.2 K is subjected to adiabatic demagnetization. If the initial magnetic field is 1 T and is reduced to 0.2 T, estimate the final temperature assuming direct proportionality between temperature and magnetic field.
In an adiabatic demagnetization process, a sample is magnetized at 2.0 K under a field of 4 T. If the field is reduced adiabatically to 0.5 T, and the magnetic heat capacity follows $C_m = A/T^2$, calculate the final temperature.
For adiabatic demagnetization obeying Curie's law and magnetic heat capacity domination, the relation is T_f^2 - $T_{f_i}nternal = T_i^2 (H_f / H_i)^2 or proportional scaling$.
Using direct field ratio for ideal paramagnet: $T_f = T_i \times (H_f / H_i)$.
Calculate the entropy change when 1 mole of a paramagnetic salt with Curie constant $C = 0.5 K·J/T^2$ is magnetized from H = 0 to H = 3 T at a constant temperature of 4 K.
Use Maxwell relation for entropy change: \Delta $S = \int_{0}^{H} (\frac{\partial m}{\partial T})_H dH$.
Since $m = C H / T$, differentiate with respect to T: (\frac{\partial m}{\partial T})_$H = - \frac{C H}{T^2}$.
Substitute values: C = 0.5, H = 3, T = 4. \Delta $S = -\frac{0.5 \times (3)^2}{2 \times (4)^2} = -\frac{0.5 \times 9}{2 \times 16} = -\frac{4.5}{32} = -0.1406 J/K$.
Answer: -0.141 J/K
Assumptions and limitations
Assumption: The paramagnetic salt obeys Curie's Law strictly (\chi \propto 1/T). Limitation: Fails at ultra-low temperatures due to magnetic saturation and dipole-dipole interactions.
Assumption: Perfect thermal isolation during demagnetization. Limitation: Parasitic heat leaks from wires and thermal radiation restrict the lowest achievable temperature.
Common misconceptions
Common error: Absolute zero can be reached by performing successive stages of adiabatic demagnetization.
Correct: Absolute zero cannot be reached in a finite number of steps as dictated by the Third Law of Thermodynamics.
As temperature approaches zero, the entropy difference between different magnetic states vanishes, rendering further cooling ineffective.
Common error: Cooling occurs because the magnetic field exerts a mechanical force on the lattice.
Correct: Cooling occurs due to the redistribution of thermal energy into the magnetic degree of freedom when magnetic entropy increases during field reduction.
The thermodynamic work is magnetic, but the temperature drop stems from entropy conservation between the spin system and the thermal lattice.
Quick revision
Adiabatic demagnetization achieves sub-Kelvin temperatures by exploiting magnetic entropy changes in paramagnetic salts.
Curie's Law of Paramagnetism: M = \frac{C \cdot B}{T}
First Law for Magnetic Systems: d$U = T dS + H dm$
Must remember
Adiabatic demagnetization is used to achieve temperatures well below $1\text{ K}$, typically down to the millikelvin range.
The technique was first proposed by P. Debye (1926) and W. Giauque (1927).
Paramagnetic salts containing ions with unpaired electrons, such as gadolinium sulfate, are utilized as working substances.
The cooling cycle consists of two primary stages: Isothermal Magnetization and Adiabatic Demagnetization.
Total entropy of the salt is the sum of lattice entropy and magnetic entropy.
Absolute zero ($0\text{ K}$) cannot be reached in a finite number of steps in accordance with the Third Law of Thermodynamics.
Residual interactions between magnetic ions and external heat leaks prevent reaching absolute zero.
Curie's Law of Paramagnetism states that magnetization $M = \frac{C \cdot B}{T}$ under high temperatures and weak magnetic fields.
Glossary
Adiabatic DemagnetizationA thermodynamic technique used to produce extremely low temperatures by rapidly reducing the magnetic field surrounding an insulated paramagnetic salt.
Magnetic EntropyThe portion of a system's entropy associated with the orientation and disorder of magnetic dipoles or spins.
Curie Constant ($C$)A material-specific constant that characterizes the magnetic susceptibility of a paramagnetic substance as a function of temperature.
Paramagnetic SaltA substance containing ions with unpaired electrons that exhibit temporary magnetization in the presence of an external magnetic field.
Schottky AnomalyA peak in the specific heat capacity of a solid at low temperatures, arising from a finite number of discrete energy levels.
Isothermal Magnetization vs Adiabatic Demagnetization
Parameter
Isothermal Magnetization (Stage 1)
Adiabatic Demagnetization (Stage 2)
Thermal Contact
In thermal contact with a constant temperature bath (e.g., liquid helium)
Thermally isolated (adiabatic system)
Magnetic Field
Strong magnetic field is applied
Magnetic field is rapidly reduced to zero
Entropy Change
Magnetic entropy decreases as dipoles align
Magnetic entropy increases while lattice entropy drops; total entropy is constant
Heat Exchange
Heat of magnetization is rejected to the surroundings
No heat exchange with the surroundings ($dQ = 0$)
Temperature Effect
Temperature remains constant ($T = \text{constant}$)
Temperature drops significantly due to energy transfer to spins
Practice Quiz
Choose an option to reveal the explanation.
1. Who independently proposed the method of adiabatic demagnetization to achieve temperatures below $1\text{ K}$?
Explanation: Adiabatic demagnetization was first proposed by P. Debye in 1926 and W. Giauque in 1927 to bypass the limitations of liquid helium pumping and reach temperatures below $1\text{ K}$.
2. Which type of substances are primarily used as working media in adiabatic demagnetization?
Explanation: Paramagnetic salts containing ions with unpaired electrons (such as gadolinium sulfate) are utilized because their magnetic dipoles can be easily aligned and randomized.
3. What is the primary purpose of Stage 1 (Isothermal Magnetization) in the cooling cycle?
Explanation: During isothermal magnetization, a strong magnetic field aligns the dipoles, lowering magnetic entropy, and the generated heat is rejected to a constant temperature bath (like liquid helium at $1\text{ K}$).
4. What happens to the magnetic entropy during Stage 2 (Adiabatic Demagnetization)?
Explanation: Since total entropy remains constant in an adiabatic process ($dS = 0$) while the lattice cools, the magnetic entropy must increase as energy is drawn from the lattice vibrations to randomize the spins.
5. According to Curie's Law of Paramagnetism, how does magnetization $M$ vary with absolute temperature $T$ under weak fields?
Explanation: Curie's Law is expressed as $M = \frac{C \cdot B}{T}$, indicating that magnetization is inversely proportional to absolute temperature $T$.
6. Why is absolute zero ($0\text{ K}$) unattainable via adiabatic demagnetization in a finite number of steps?
Explanation: The Third Law of Thermodynamics states that the entropy of a pure crystalline substance approaches zero as temperature approaches absolute zero, making it impossible to reach $0\text{ K}$ in a finite number of steps.
7. What is the Maxwell thermodynamic relation connecting entropy $S$, magnetic field $H$, and magnetic moment $m$?
Explanation: Derived from the magnetic internal energy differential $dU = TdS + Hdm$ or corresponding thermodynamic potentials, the Maxwell relation is $\left(\frac{\partial S}{\partial H}\right)_T = \left(\frac{\partial m}{\partial T}\right)_H$.
8. What is the formula for the magnetic heat capacity $C_m$ of a paramagnetic salt in the low-temperature limit ($T \ll 1\text{ K}$)?
Explanation: At extremely low temperatures, the magnetic heat capacity follows a $T^{-2}$ dependence, expressed as $C_m = \frac{A}{T^2}$ (associated with the Schottky anomaly).
9. In the adiabatic temperature drop relation $T_f = T_i \left( \frac{H_f}{H_i} \right)^n$, what does $n$ typically represent for an ideal paramagnetic substance?
Explanation: For an ideal paramagnetic substance obeying Curie's law, the temperature drop during adiabatic demagnetization from initial field $H_i$ to final field $H_f$ follows an exponent of $n = 1/2$ (or depends on the specific Curie-Weiss parameters).
10. What prevents complete disorder of magnetic dipoles at $0\text{ K}$ even when the external magnetic field is zero?
Explanation: Residual internal interactions (such as dipole-dipole or exchange interactions between magnetic ions) create internal molecular fields that prevent complete randomization.
11. What is the expression for work done in an isothermal magnetization process of a magnetic system?
Explanation: The magnetic work done on the system during magnetization is given by $W = - \int m \cdot dH$, where $m$ is the magnetic moment and $H$ is the applied field.
12. What constitutes the total entropy $S$ of a paramagnetic salt at low temperatures?
Explanation: The total entropy of a paramagnetic salt is composed of lattice entropy (due to atomic vibrations) and magnetic entropy (due to spin alignments).
13. Why does lattice entropy become negligible compared to magnetic entropy at extremely low temperatures?
Explanation: According to Debye's $T^3$ law, lattice heat capacity and lattice entropy decrease rapidly as $T^3$, becoming negligible compared to magnetic entropy at temperatures below $1\text{ K}$.
14. What is the First Law of Thermodynamics expression for a magnetic system undergoing reversible changes?
Explanation: For a magnetic system, the internal energy change is given by $dU = T dS + H dm$ (or $dU = T dS + \mu_0 V H dM$ depending on the convention used for magnetic variables).
15. Which factor imposes a lower bound on the minimum attainable temperature in a practical adiabatic demagnetization setup?
Explanation: Residual heat leaks from the surroundings and experimental apparatus introduce thermal energy, setting a practical lower limit on the minimum temperature achieved.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. In the production of low temperatures by adiabatic demagnetization, what is the thermodynamic condition of the paramagnetic salt during Stage 2?
Explanation: Stage 2 is adiabatic demagnetization, meaning the system is thermally isolated from its surroundings so that no heat enters or leaves ($dQ = 0$), keeping entropy constant ($dS = 0$).
2. Which of the following Maxwell's relations is correctly formulated for a magnetic system where $S$ is entropy, $H$ is magnetic field, and $m$ is magnetic moment?
Explanation: From the thermodynamic potential for magnetic systems, the Maxwell relation derived is $\left(\frac{\partial S}{\partial H}\right)_T = \left(\frac{\partial m}{\partial T}\right)_H$.
3. Why does a paramagnetic salt cool down when its surrounding magnetic field is abruptly reduced to zero adiabatically?
Explanation: When the field is removed adiabatically, total entropy is conserved. To compensate for the drop in field, magnetic entropy increases by drawing thermal energy from the lattice, thereby cooling the salt.
4. According to the Third Law of Thermodynamics, what happens to the entropy of a pure crystalline substance as absolute zero ($T \to 0$) is approached?
Explanation: The Third Law of Thermodynamics states that $\lim_{T \to 0} S = 0$ for any pure crystalline substance in internal equilibrium.
5. In the low-temperature limit ($T \ll 1\text{ K}$), the magnetic heat capacity $C_m$ of a paramagnetic salt varies with absolute temperature $T$ as:
Explanation: Due to internal crystal fields and dipole interactions (Schottky anomaly), the magnetic heat capacity at very low temperatures follows $C_m = \frac{A}{T^2}$, meaning $C_m \propto T^{-2}$.
6. Which of the following statements correctly identifies a limitation in reaching absolute zero?
Explanation: Residual internal molecular fields prevent complete thermal randomization or ordering, and the Third Law forbids reaching $0\text{ K}$ in a finite number of steps.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. What is the primary role of the liquid helium bath at $1\text{ K}$ during Stage 1 of adiabatic demagnetization?
Explanation: During isothermal magnetization, heat is generated as dipoles align; the liquid helium bath acts as a thermal reservoir to absorb this rejected heat.
2. If an ideal paramagnetic salt obeys Curie's law, how does the final temperature $T_f$ depend on the initial temperature $T_i$ and magnetic fields $H_i, H_f$?
Explanation: For an ideal paramagnetic substance obeying Curie's law, the adiabatic temperature drop relation is $T_f = T_i \left(\frac{H_f}{H_i}\right)^{1/2}$ (with $n = 1/2$).
3. Which thermodynamic potential's natural variables are temperature and magnetic field for magnetic systems?
Explanation: For magnetic systems, the thermodynamic potential analogous to Gibbs free energy is $G = U - TS - mH$, making $T$ and $H$ natural variables.
4. What is the physical origin of magnetic entropy in a paramagnetic salt?
Explanation: Magnetic entropy arises from the orientational disorder of the magnetic dipoles associated with unpaired electrons in the paramagnetic ions.
5. In the context of adiabatic demagnetization, what happens to the lattice vibrations when thermal energy is transferred to the spin system?
Explanation: As thermal energy is drawn from the lattice reservoir to randomize the magnetic spins, the lattice vibrations decrease, leading to a net drop in temperature.
6. Why cannot absolute zero be reached in a finite number of steps according to thermodynamics?
Explanation: As $T \to 0$, the entropy difference between magnetized and demagnetized states vanishes (Third Law), meaning an adiabatic step produces negligible temperature reduction.
7. Which of the following equations represents the First Law for a magnetic system including magnetic work?
Explanation: The First Law combined with reversible magnetic work is given by $dU = TdS + Hdm$ (or $TdS - mdH$ depending on choice of work conjugate).
8. What is the typical temperature range achieved using standard adiabatic demagnetization with paramagnetic salts like gadolinium sulfate?
Explanation: Adiabatic demagnetization typically takes over where liquid helium pumping leaves off ($1\text{ K}$), reaching down into the millikelvin range ($1\text{ mK}$ to $1\text{ K}$).
9. How does the entropy of a paramagnetic salt vary with an applied magnetic field at constant temperature?
Explanation: Applying a strong magnetic field aligns the dipoles, ordering the magnetic system and thereby decreasing the magnetic entropy.
10. Which scientist is credited with first pointing out the utility of adiabatic demagnetization in achieving ultra-low temperatures in 1926?
Explanation: Peter Debye first proposed the method independently in 1926, followed closely by William Giauque in 1927 (who also successfully performed the experiment).