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Schottky anomalous heat capacity $C$ for a two-level system. $T$ is the absolute temperature.

The Schottky anomaly, named after Walter H. Schottky, is an effect observed in solid-state physics where the specific heat capacity of a solid at low temperature has a peak. It is called anomalous because the heat capacity usually increases with temperature, or stays constant. In systems with a limited number of energy levels at low temperature, U(T) changes rapidly as the higher energy levels become thermally accessible. Since the specific heat is how the energy changes with temperature ($C={\frac {dU}{dT}}$), it will peak when the change in energy is large, i.e. when the new energy levels become accessible.

This effect can be explained from multiple viewpoints. As stated above the heat capacity is defined as how the systems internal energy (U) changes as a function of temperature ($C={\frac {dU}{dT}}$). At T = 0, all particles are in the state with lowest energy, the ground state, and the internal energy is constant. As temperature increases, thermal fluctuations begin to occur and a few particles can transition to the state with higher energy. This causes the internal energy to change abruptly, and thus the heat capacity to peak. As temperature continues to increase both levels become equally populated. This stabilizes the internal energy and the heat capacity accordingly dies off.

This effect persists in systems with more than two energy levels at low temperature. As the lowest two energy levels become equally populated, the internal energy levels off. As T continues to increases, another energy level becomes accessible, and one expects a peak in heat capacity corresponding to the third energy level being "thawed" out. Scientists can use this phenomena to probe the low temperature energy landscape of materials, typically magnetic solids.

This effect can also be explained by looking at the change in entropy of the system. At zero temperature only the lowest energy level is occupied, entropy is zero, and there is very little probability of a transition to a higher energy level. As the temperature increases, there is an increase in entropy and thus the probability of a transition goes up. As the temperature approaches the difference between the energy levels there is a broad peak in the specific heat corresponding to a large change in entropy for a small change in temperature. At high temperatures all of the levels are populated evenly, so there is again little change in entropy for small changes in temperature, and thus a lower specific heat capacity.

Heat capacity and internal energy of two level system as a function of temperature. Energy has been scaled by a factor of 1/2 for clarity

Here we can show the anomaly in a simple toy model of a system with two energy levels,$-k_{B}\Delta \,$ and $k_{B}\Delta \,$, at low temperature.

The partition function reads,

$Z(T)=e^{\Delta /T}+e^{-\Delta /T}\,$

Then the internal energy,$U$,

$U=k_{B}T^{2}{\frac {\partial \ln Z}{\partial T}}={\frac {\Delta -\Delta e^{\frac {2\Delta }{T}}}{e^{\frac {2\Delta }{T}}+1}}$

Finally the heat capacity $C$,

$C={\frac {dU}{dT}}={\frac {4\Delta ^{2}e^{\frac {2\Delta }{T}}}{T^{2}\left(e^{\frac {2\Delta }{T}}+1\right)^{2}}}$

Note how the internal energy is initially constant, as all the particles are in the ground state. Once the particles have enough thermal energy, they can start transitioning to the higher level, thus changing the energy of the system. This corresponds to the jump in the heat capacity, the Schottky Anomaly. At high T, the two states are equally populated and the energy stabilizes again.

As stated early, the above description of a two level system is a simple one. In real systems, particles not only act with the external field but with each other. Therefor coupling between particles must be taken into account. This is a more physical description of materials, namely magnetic materials, which posses spin degrees of freedom which interact with each other and the applied external field. The Ising model accomplishes this goal. The model is hard to solve but does possess solutions in 1D.

Heat capacity and internal energy of 1D Ising Model as a function of temperature at H = 1. Energy has been scaled by a factor of 1/2 for clarity. Schottky peaks shifts but is still clear

The partition function reads, where H is the external field and J parameterizes the coupling, the energy levels associated with H are $\Delta =\pm 1$

$Z(T)=e^{J/T}\left({\sqrt {\sinh ^{2}\left({\frac {H}{T}}\right)+e^{-{\frac {4J}{T}}}}}+\cosh \left({\frac {H}{T}}\right)\right)\,$

Note when J=0, there is no coupling and you recover the partition function for the two state system with $H\sim \Delta \,$ ,

$Z(T)|_{J=0}={\sqrt {\cosh ^{2}\left({\frac {H}{T}}\right)}}+\cosh \left({\frac {H}{T}}\right)=2\cosh \left({\frac {H}{T}}\right)=e^{H/T}+e^{-H/T}\,$

Now onto the desired quantities. First the energy,

$U=k_{B}T^{2}{\frac {\partial \ln Z}{\partial T}}={\frac {{\frac {e^{-{\frac {4J}{T}}}\left(2J-{\frac {1}{2}}He^{\frac {4J}{T}}\sinh \left({\frac {2H}{T}}\right)\right)}{\sqrt {\sinh ^{2}\left({\frac {H}{T}}\right)+e^{-{\frac {4J}{T}}}}}}-H\sinh \left({\frac {H}{T}}\right)}{{\sqrt {\sinh ^{2}\left({\frac {H}{T}}\right)+e^{-{\frac {4J}{T}}}}}+\cosh \left({\frac {H}{T}}\right)}}-J$

Then the heat capacity, whose expression is too large to fit on the page. The plots of the two quantities are shown on the right. The Schottky anomaly is similarly observed in the 1D Ising model suggesting the phenomenon does exist in real systems. The same qualitative behavior is observed as the toy model. The lower ground state energy is attributed to the particles being aligned with the field and all parallel (spin up). This sheds light on the power and success of Statistical Mechanics as a simple toy model can predict a low temperature phenomenon. This also allows one to see how quickly interactions make a model unsolvable. In one dimension the Ising model is difficult, in just 2D it quickly becomes intractable.

Heat capacity and internal energy of three level system as a function of temperature. Energy has been scaled by a factor of 1/2 for clarity. Second Schottky peak arises due to third energy level thawing out

$C={\frac {dU}{dT}}$

As stated above, one expects to see multiple Schottky peaks if there are more than 2 levels available at low temperature. This displays how the Schottky anomaly can be used as an experimental probe .

Heat capacity of three level, 1D Ising Model as function of temperature and external field

Heat capacity and internal energy of three level, 1D Ising Model as a function of temperature at H = 0.9. Energy has been scaled by a factor of 1/2 for clarity. Two Schottky peaks are visible

First we will show how two peaks arise in a 3 level toy model. Then similarly demonstrate the peaks arising in a 3 level Ising model for a particular value of field. For the three level system we have energy levels, ,$-k_{B}\Delta \ ,\ 0\,$ and $-10k_{B}\Delta \,$. The third level must be far enough away energetically from the first two levels such that its peak does not overlap with the first peak. The partition function reads,

$Z(T)=1+e^{\Delta /T}+e^{-10\Delta /T}\,$

The energy and heat capacity can be computed from this but will be suppressed for the sake of brevity. The plots of the two quantities are shown to the right. One sees the initial peak in the heat capacity signifying the second level(Δ=0) beginning to be populated. As T continues to increases, all of the sudden the particles can access a new level (Δ=10) and the heat capacity jumps again.

Now we will show the two Schottky peaks arising in the more realistic, but mathematically formidable 3 level Ising model in 1D, or Potts model. The energy levels associated with H are $\Delta =\pm 1\ ,\ 0$. The partition function can be obtained via the Transfer-matrix method,

$Z(T)={\frac {1}{2}}\left({\sqrt {\left(-e^{\frac {-H-J}{2T}}+e^{\frac {H+J}{T}}-e^{\frac {2J-H}{2T}}\right)^{2}+8e^{\frac {10H-2J}{2T}}}}+e^{\frac {-H-J}{2T}}+e^{\frac {H+J}{T}}+e^{\frac {2J-H}{2T}}\right)\,$

As before, due to the nature of the partition function, the expressions for the energy and heat capacity will be suppressed. As seen in the 3D plot of the heat capacity the landscape is rich. Two Schottky peaks are visible at both H = -1 and H ~ 1. For the sake of continuity we inspect the peaks around H = 1. Here we see two Schottky peaks, associated with the two energy levels above the ground state. Here in the physical model of a three level system with coupling, the spacing of the energy levels need not be so largely spaced as in the toy model for both peaks to become visible. This suggests that the Schottky anomaly in real systems has high resolution, being able to resolve closely spaced energy levels. The external field also adds another experimental knob as one can move the peaks in order to be better resolve them or make new ones appear, hinting at closely spaced energy levels being pushed apart by the field.

This anomaly is usually seen in paramagnetic salts or even ordinary glass (due to paramagnetic iron impurities) at low temperature. At high temperature the paramagnetic spins have many spin states available, but at low temperatures some of the spin states are "frozen out" (having too high energy due to crystal field splitting), and the entropy per impurity is lowered.

References

Category:Thermodynamic properties Category:Condensed matter physics