Schottky anomaly

The Schottky anomaly (after Walter Schottky ) is in statistical physics a peak of the heat capacity at low temperatures in two-level systems . The Schottky anomaly can be used to determine energy levels in solids .

Derivation

Course of the heat capacity with Schottky anomaly

In a system whose particles can either take on the energy 0 or (two-level system), the expected value of the energy of a particle in the canonical ensemble is : ${\displaystyle \epsilon }$

${\displaystyle \langle \epsilon \rangle =\epsilon \cdot {\frac {e^{-\beta \epsilon }}{1+e^{-\beta \epsilon }}}={\frac {\epsilon }{e^{+\beta \epsilon }+1}}}$

with the inverse temperature and the Boltzmann constant . The total energy of independent particles is thus: ${\displaystyle \beta ={\frac {1}{k_{\mathrm {B} }T}}}$ ${\displaystyle k_{\mathrm {B} }}$ ${\displaystyle N}$

${\displaystyle U=N\langle \epsilon \rangle ={\frac {N\epsilon }{e^{+\beta \epsilon }+1}}}$

The heat capacity is therefore:

${\displaystyle C=\left({\frac {\partial U}{\partial T}}\right)=-{\frac {1}{k_{\mathrm {B} }T^{2}}}{\frac {\partial U}{\partial \beta }}=Nk_{\mathrm {B} }\left({\frac {\epsilon }{k_{\mathrm {B} }T}}\right)^{2}{\frac {e^{+{\frac {\epsilon }{k_{\mathrm {B} }T}}}}{\left(e^{+{\frac {\epsilon }{k_{\mathrm {B} }T}}}+1\right)^{2}}}}$

This function is shown as a function of the temperature in the diagram on the right. The clear peak at is called the Schottky anomaly. At temperatures as prevailing in the anomaly, it is possible to excite many particles (e.g. spins in a magnet) to the upper energy level, while below this temperature too little energy is available for many excitations and the system at significantly higher temperatures goes into saturation. In contrast to phase transitions in which the heat capacity diverges, the peak due to the Schottky anomaly is broad and smooth. ${\displaystyle k_{\mathrm {B} }T\approx 0{,}417\epsilon }$

Individual evidence

1. ^ Rudolf Gross: Physics IV - Atoms, Molecules, Heat Statistics. (PDF) Lecture notes for the lecture in SS 2003. Walther Meissner Institute, Bavarian Academy of Sciences, Chair for Technical Physics (E23) at the Technical University of Munich, 2003, p. 461 , accessed on March 31, 2015 (lecture notes). 
2. ↑ Charles Kittel: Introduction to Solid State Physics . Oldenbourg Verlag, New York 2006, ISBN 3-486-57723-9 , pp. 350 .
3. ↑ Stephen Blundell: Magnetism in Condensed Matter . Oxford University Press, New York 2001, ISBN 0-19-850591-4 , pp. 27 .