A paramagnetic salt is cooled and its heat capacity per ion is plotted against temperature

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Statistical foundations and the equilibrium ensembles · Statistical foundations and the equilibrium ensembles

Thermal Physics, Ch Two-level systems, paramagnets and the Einstein solid

Problem

A paramagnetic salt is cooled and its heat capacity per ion is plotted against temperature. Each ion is a two-level system with a gap

\varepsilon

, the lower level set at zero, and the ions are independent. The following values are read off the plotted curve, whose vertical axis is in units of

k_B

per ion:the curve rises from zero at the lowest temperatures;it has a single maximum of height

0.44\,k_B

T = 1.6\,\mathrm{K}

;it has fallen below

0.03\,k_B

everywhere above

12\,\mathrm{K}

. A second graph on the same page sketches the entropy per ion,

S

, against the energy per ion,

U

; it is an arch running from

U = 0

U = \varepsilon

with its top at

U = \varepsilon/2

. ((a)) From the position of the maximum, find

\varepsilon/k_B

in kelvin and

\varepsilon

in joules. ((b)) Express the height of the maximum as a heat capacity per mole of ions. ((c)) A pumping technique prepares the same ions with

U = 0.70\,\varepsilon

per ion. Say from the sketch what the sign of the temperature is there, compute that temperature, and state whether the prepared sample is hotter or colder than the

1.6\,\mathrm{K}

bath.

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