The heat capacity that goes up and comes down

Half a kT for every way of moving found that a heat capacity is a count: every quadratic term in a system’s energy holds half of $kT$ at temperature $T$, so a monatomic gas holds $\tfrac{3}{2}k$ per atom, a diatomic one $\tfrac{5}{2}k$ once its rotations are awake, and a crystal $3k$ per atom. The share that is not half a kT sharpened the rule: it is not about degrees of freedom but about how the energy depends on each coordinate, and an energy linear in its coordinate holds a whole $kT$ rather than half.

Both essays also found where the count fails at low temperature. A quantum degree of freedom whose levels are spaced by much more than $kT$ cannot be excited by the thermal motion around it, and it is frozen out: hydrogen’s rotations below a hundred kelvin, its vibrations below a few thousand. As the temperature rises past the spacing, each frozen degree of freedom wakes and climbs to its equipartition share.

That climb assumes something not always true: that the stack of levels goes on for ever. A harmonic oscillator’s does, with levels evenly spaced as far up as one cares to look. A great many real degrees of freedom have only a few levels — an electron spin in a magnetic field has two, an ion in a crystal may have a handful split off from its ground state, a nucleus with spin has a small set in the crystal’s field. For those, the story is not a climb to a plateau. It is a rise and a fall.

Two levels, and a hump #

Take the simplest case, a system with two levels a gap $\Delta$ apart. At temperature $T$ the probability of finding it in the upper level relative to the lower is the Boltzmann factor $e^{-\Delta/kT}$, and its average energy is

$$\langle E\rangle = \frac{\Delta}{1 + e^{\Delta/kT}}.$$

At low temperature almost every system is in the lower level and the energy is nearly zero. At high temperature the two levels are nearly equally occupied and the energy approaches $\Delta/2$ — and can go no higher, because there is no third level. The heat capacity is the slope of the energy against temperature,

$$\frac{C}{k} = \left(\frac{\Delta}{kT}\right)^2\frac{e^{\Delta/kT}}{\left(1 + e^{\Delta/kT}\right)^2},$$

and a function that starts flat at zero, rises, and flattens again at $\Delta/2$ has a slope that starts at zero, rises to a maximum, and falls back.

The peak sits at $kT = 0.417\Delta$, where the heat capacity is $0.439k$ — less than half of what the oscillator reaches. Below it, the two systems behave alike: both are frozen, and both thaw along the same exponential, since at low temperature only the first excited level matters and the two systems have the same one. Above the peak they part company. The oscillator, having more levels above, goes on absorbing heat into them; the two-level system, with both levels already nearly equal, cannot. By $kT = 3\Delta$ its heat capacity has fallen to three per cent of a $k$.

This is called the Schottky anomaly, after Walter Schottky, who described it in 1922 to account for heat capacities that refused to settle. It is anomalous only against equipartition’s expectation. As a piece of statistical mechanics it is the plainest possible result: the heat a system can hold is bounded by the energy its levels allow.

Why equipartition needed an unbounded energy #

The comparison shows precisely what equipartition assumes. The derivation that gives half a $kT$ per quadratic term integrates the Boltzmann factor over a coordinate running from minus infinity to infinity; the half comes from the Gaussian integral, and the infinite range is essential. A coordinate whose energy is bounded — an angle in a potential with a maximum, a spin with two orientations, a level structure that ends — does not satisfy the derivation’s hypothesis, and has no share to claim. The bound need not be quantum. A methyl group on a molecule can twist about its bond against a potential with a few hills, and classically, at low temperature, it rocks in one valley like an oscillator, holding $k$ — half a $kT$ for its kinetic energy and half for its potential. Hot enough to clear the hills, it rotates freely, its potential energy is bounded by the height of the hills and averages to a constant, and its heat capacity falls to the $k/2$ of its kinetic part alone. The potential’s share rose to its equipartition value and then fell away, for the same reason as the hump: a bounded energy can absorb only so much.

Stacks of more levels interpolate between the two cases. A stack of three levels peaks higher and later than two; five, higher and later still; ten, at almost the full $k$ and at a temperature near its spacing. Each follows the oscillator closely until the temperature is high enough to reach its top level, and then turns over. In the limit of infinitely many levels the turnover moves to infinite temperature and the curve becomes the oscillator’s, with equipartition’s $k$ as its plateau.

So the oscillator’s plateau is not a general law that two-level systems violate. It is the special case of a stack with no top. Equipartition is a statement about the high-temperature limit of systems whose energy is unbounded, and quantum mechanics supplies, in finite level structures, the systems for which that limit has nothing to say.

The fluctuation reading #

There is a second way to see the hump, through the connection the temperature a molecule does not have found between heat capacity and the size of a system’s energy fluctuations: $C = \langle(\Delta E)^2\rangle/kT^2$. A system that can hold more heat is one whose energy wanders more at a given temperature.

A two-level system’s energy can take only two values, $0$ and $\Delta$, so its fluctuations are those of a coin flip: largest when the two outcomes are equally likely, and small when either dominates. At low temperature the system is almost always in the lower level and barely fluctuates. At high temperature the two outcomes are nearly equally likely and the fluctuation in energy is as large as it can be, $\Delta/2$ — but it stays at that value as the temperature rises further, while the $kT^2$ in the denominator grows without limit. The heat capacity therefore falls as $1/T^2$. The peak is where the fluctuations have nearly reached their ceiling and the temperature is still small enough not to swamp them.

Past the top level #

The two-level system has one more property the oscillator lacks, and it shows what the falling side of the hump means. Its energy cannot exceed $\Delta$, and at infinite temperature it is $\Delta/2$. Between those, with more systems in the upper level than the lower, it is in a state no positive temperature produces — the population inversion hotter than any temperature there is assigned a negative temperature. The heat capacity, read as the energy’s slope against $1/T$ rather than against $T$, is continuous through infinite temperature into that negative region, and the hump has a mirror image on the other side.

The falling side of the hump is therefore not the system losing interest in heat. It is the system approaching the middle of its range, where adding energy changes its entropy least, so that each unit of energy changes its temperature most. The same bounded energy that gives the hump makes negative temperatures possible, and the oscillator, with no top to its stack, has neither.

The entropy under the hump #

The heat absorbed by the two-level system on the way up has a definite total. Integrating the heat capacity divided by temperature from absolute zero to infinity gives the total entropy gained, and for any system with $N$ levels that is

$$\int_0^\infty \frac{C}{T}\,dT = k\ln N,$$

the logarithm of the number of states, which is all the entropy a system with $N$ equally occupied states can have.

That makes the hump a counting instrument. A measured heat-capacity peak, integrated over temperature, says how many states the degree of freedom behind it has, without saying anything about what they are. A law about spectra, not about heat found the same arithmetic in the third law: the entropy of a system in its ground state is $k$ times the logarithm of the ground state’s degeneracy, so a ground state split into $N$ levels by some small interaction carries $k\ln N$ of entropy down to whatever temperature that splitting allows — and releases it, through a Schottky hump, as the temperature falls below the splitting. The entropy still there at zero in ice is a case where no such splitting was ever reached, so the hump never happened.

A hump a magnet can move #

The two-level system that is easiest to control is an electron spin in a magnetic field. A dilute paramagnetic salt — a crystal in which a few of the ions carry unpaired electrons, well separated so that they barely feel each other — has, in a field $B$, two levels for each such ion, split by $g\mu_B B$, with $g \approx 2$ and $\mu_B$ the Bohr magneton. The gap is about 1.34 kelvin per tesla in temperature units.

The hump keeps its height and its shape, because those are fixed by there being two levels; only the temperature scale moves, in proportion to the field. Doubling the field doubles the temperature of the peak. That is the property used in the oldest method of reaching temperatures below a kelvin, adiabatic demagnetisation, which the staircase that never reaches the floor described: magnetise the salt at a few kelvin, where the field’s splitting is large compared with $kT$ and the spins are ordered, insulate it, and reduce the field. The entropy of the spins cannot change while the salt is isolated, and since the entropy depends only on the ratio of gap to temperature, reducing the gap tenfold reduces the temperature tenfold. The salt’s Schottky hump is the reservoir of entropy that makes the trick work, and its heat capacity at the final temperature decides how much heat the cooled salt can absorb before it warms up.

Telling a hump from a transition #

A peak in a measured heat capacity is the commonest signal that something interesting happens in a material, and the commonest cause is a phase transition: a magnet ordering, a crystal changing structure, a superconductor forming. A Schottky hump is a peak with no transition behind it, and three features tell them apart. Its shape is fixed: a two-level hump has a definite ratio of width to position, broad on the high-temperature side, with no sharp top, whereas a transition’s peak sharpens as the sample improves and, at a continuous transition, has a cusp or a divergence. Its position moves in proportion to whatever sets the gap — linearly with a magnetic field for a spin — whereas a transition temperature usually shifts much less. And its area is a logarithm of a small whole number, which a transition’s need not be. A measured peak that passes all three tests is a set of levels, counted.

Where the hump is found #

Against a crystal’s vibrations. Any real solid also has a lattice heat capacity, rising from zero as $T^3$ at low temperature, as how many ways there are to vibrate derived, and reaching $3k$ per atom at high temperature. A Schottky hump stands out against it only in the window where it is larger.

Because the lattice term grows as $T^3$ and the hump’s tail falls as $1/T^2$, the window is narrow, and it lies at low temperature. Schottky humps are found in rare-earth compounds, where an ion’s electrons are split by the crystal’s electric field into a few levels a few to a few tens of kelvin apart; in magnetic insulators below their ordering temperature; and, at millikelvin temperatures, in metals whose nuclei carry spin, split by the magnetic field the nuclei feel from their own electrons. In each case the hump’s position measures a splitting and its area counts states, and the measurement is used to identify which levels an ion has.

In metals cooled to the lowest temperatures. Copper, used to build the coldest refrigerators because its nuclear spins can themselves be demagnetised, has a nuclear Schottky hump in any applied field, at microkelvin temperatures. It is the reservoir of entropy those refrigerators exploit, and at the same time a heat load: a copper stage warmed by a stray microwatt must pass through its own hump, absorbing heat as its nuclear spins disorder, before its temperature can rise.

In glasses. Amorphous solids have a heat capacity at very low temperature that is linear in $T$ rather than cubic, which was a puzzle until 1972, when Anderson, Halperin and Varma and, independently, Phillips proposed that a glass contains a broad distribution of two-level systems — atoms or groups of atoms that can sit in either of two nearly equivalent positions and tunnel between them. Each contributes its own small Schottky hump, and the glass, having fallen out of equilibrium as it cooled in the way the entropy that depends on how fast it was cooled described, keeps a frozen-in spread of them; summed over a broad distribution of gaps, the humps add to a capacity proportional to temperature. The same two-level systems are what the circuit that forgets its charge found absorbing energy from superconducting qubits in the oxide on their surfaces, and limiting how long the qubits remember their state.

What the simple picture leaves out #

The systems are independent. Each two-level system is assumed to feel only the applied field and its own gap. Spins close enough to feel each other’s fields interact, and at low enough temperature they order collectively, with a sharp transition instead of a smooth hump. Dilution delays it; nothing removes it.

The gap is sharp. Real gaps are broadened by strain, by neighbouring defects and by the spread of local fields in a crystal. A distribution of gaps spreads the hump out, and a broad enough one changes its shape completely, as in a glass.

Two levels are often really more. An ion with a larger spin has more levels in a field, and a crystal field can split them unevenly. The hump then has a different height and shape, and sometimes two humps, one for each group of levels. The area under them still counts the total number of states.

Still open: what the two-level systems in a glass actually are #

The tunnelling model of glasses has been remarkably successful. It explains the linear heat capacity, a thermal conductivity rising as $T^2$, a characteristic absorption of sound and microwaves that saturates at high intensity, and a dozen other low-temperature properties shared by nearly every amorphous solid, from window glass to amorphous silicon to polymers. Its central assumption is a broad, nearly flat distribution of two-level systems with a density of about $10^{45}$ per joule per cubic metre, roughly the same in all glasses.

What the two-level systems are, physically, has never been settled. In some glasses they can be identified with specific groups of atoms that rotate or shift between two positions; in most they cannot. Why their density is nearly universal, varying by less than a factor of ten across materials whose chemistry differs completely, is the deeper puzzle, and suggests that the two-level systems are a collective property of disorder rather than a feature of any particular atom. The question has practical weight now that the same defects limit superconducting quantum computers, and whether glasses can be made with far fewer of them — by slow deposition, by growing them on heated substrates — is being actively explored.

The heat-capacity argument is the part that is certain. A degree of freedom with a ceiling cannot hold equipartition’s share. It holds less, and only over a window of temperature; it gives up its heat as the temperature drops through its gap; and the entropy it releases on the way down is the logarithm of its number of states. Every Schottky hump ever measured is that statement, read off a calorimeter.