به انگلیسی

Order appears, symmetry goes

Science · Thermal & Statistical Physics · ages 22-24

Name ______________________ Date ____________

Which order parameter measures spin alignment in a ferromagnet?
- Electrical resistance
- Magnetisation
- Thermal conductivity
What is the mathematical signature of a thermodynamic phase transition?
- Nonanalytic free energy in the thermodynamic limit
- A discontinuity in the free energy at every transition
- A divergent heat capacity at every possible transition
Every continuous phase transition must have a divergent heat capacity.
Circle one: True False
At coexistence at 300 K, heating changes the molar entropy by +0.2 J/(mol K). How much latent heat is absorbed per mole?
- 300 J/mol
- 60 J/mol
- 150 J/mol
As a field crosses a coexistence value, magnetisation jumps but entropy stays continuous. How is the transition classified?
- Continuous, because entropy has no jump
- No transition, because latent heat is zero
- First-order, because magnetisation jumps
A zero-field Ising model has spins s_i = ±1. Selecting a state with positive mean spin breaks which symmetry?
- Global spin reversal: s_i → −s_i
- Continuous rotation of the scalar spins
- Changing every interaction strength J
An equilibrium average for a finite zero-field Ising sample is zero below its ordering temperature. What can explain this without ruling out order?
- A zero mean proves that every spin stays unaligned at all times
- An equal mixture of opposite ordered states can average to zero
- A zero mean proves the energy is independent of every spin
A nonzero field favours positive Ising spins. A student calls the resulting magnetisation proof of spontaneous symmetry breaking. What is missing?
- The field leaves H symmetric and selects a spontaneous state
- The field breaks the symmetry of H explicitly
- The field prevents a finite sample from becoming magnetised

Answer key

For grown-ups. Fold this page away before handing over the rest.

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Magnetisation · Magnetisation measures the net alignment of magnetic moments. Resistance and thermal conductivity do not directly measure that alignment.
Nonanalytic free energy in the thermodynamic limit · The defining feature is nonanalytic free energy in the thermodynamic limit. Neither a jump in free energy itself nor divergent heat capacity is required.
False · Heat capacity can have a finite jump at a continuous transition. Continuous first derivatives do not require every higher derivative to diverge.
60 J/mol · Use L = TΔs. Multiplying 300 K by 0.2 J/(mol K) gives 60 J/mol absorbed on heating.
First-order, because magnetisation jumps · With other variables fixed, magnetisation is minus the first derivative of free energy with respect to its conjugate field. A magnetisation jump therefore means that derivative jumps, making the transition first-order even without an entropy jump.
Global spin reversal: s_i → −s_i · Reversing every spin leaves the zero-field energy unchanged but changes positive mean spin into negative mean spin. Selecting one state breaks this discrete symmetry.
An equal mixture of opposite ordered states can average to zero · Positive and negative ordered configurations can be equally weighted. Their magnetisations cancel in the average even though each configuration has ordered correlations.
The field breaks the symmetry of H explicitly · A selecting field already distinguishes positive and negative spins in the energy. Magnetisation in that field alone does not demonstrate spontaneous breaking.