GATE 2024 PH – Question 44

Thermodynamics and Statistical Mechanics · macrostates and microstates · 2 marks · Multiple select

An infinite one-dimensional lattice has spin-half ions at every site, pointing only along +z or −z. Denote entropies of paramagnetic, ferromagnetic and antiferromagnetic configurations by $S_P,S_F,S_A$. Which relation(s) is/are true?

$S_P>S_F$
$S_A>S_F$
$S_A=4S_F$
$S_P>S_A$

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Show answer and explanation

Correct answer: (A) $S_P>S_F$; (D) $S_P>S_A$

Explanation

There are $N$ sites, each with spin along $+z$ or $-z$. Entropy is $S=k_B\ln\Omega$, where $\Omega$ is the number of configurations of the phase being considered.

**Paramagnetic** state: spins are free and independent, so all $2^N$ configurations are possible: $S_P=k_BN\ln2$.
**Ferromagnetic** state: all spins aligned. There are only 2 configurations (all up or all down): $S_F=k_B\ln2$.
**Antiferromagnetic** state: spins alternate. There are only 2 configurations (up-down-up... or down-up-down...): $S_A=k_B\ln2$.

Compare:
- **A. $S_P>S_F$.** True. ✓
- **B. $S_A>S_F$.** False; they are equal.
- **C. $S_A=4S_F$.** False.
- **D. $S_P>S_A$.** True. ✓

Answer **A and D**.