FR38 · Heat transport

Transient Thermal Grating: When Heat Rings Like Sound

Paint a striped temperature pattern with two laser pulses. Does it fade like diffusion, or swing hot and cold like a wave? Usually hot and cold stripes just fade, faster when closer together: that is diffusion. When collisions that keep the phonons' total momentum far outnumber those that destroy it, heat travels as a wave called second sound and the stripes swap, so for a moment the hot stripes are the colder ones. Freely flying phonons cause a small swap too; the depth and timing of the swap tell them apart.

Two pump pulses paint hot and cold stripes. Watch them fade by diffusion, ring by free flight, or swap sign as second sound, then test which model fits held-out data.

Grating period $10\,\mathrm{\mu m}$

Lowest grating amplitude (below zero: the stripes swapped) $-0.876$

Diffusivity from an exponential fit, over the Fourier value $0.025$

Time after the pump pulses $1.59\,\mathrm{ns}$

Set the grating and the crystal

Grating period (µm)

10

L

Resistive scattering time (ns)

5

\tau_R

Normal scattering time (ns)

0.01

\tau_N

Drag the stage sideways to change the period.

Detector noise and seed

Detector noise, as a fraction of the starting signal

0.02

Seed of the detector noise

3802

Model, units and sources

$\frac{\partial g}{\partial t}+i q v\mu\,g=-\frac{g-\theta}{\tau_R}-\frac{g-\theta-3\mu J}{\tau_N},\qquad \theta=\tfrac12\!\int g\,d\mu,\quad J=\tfrac12\!\int \mu\,g\,d\mu$

One gray phonon branch with an assumed speed of 10 µm/ns and two assumed scattering times. The grating amplitude comes from a 64-direction solver with exact streaming; a separate Laplace-transform solution of the same equation checks it. The scene magnifies the pump crossing angle tenfold and is a schematic of the geometry, not graphite.

Huberman et al. (2019) · Maznev, Johnson and Nelson (2011) · Lee et al. (2015)

Course chapter: reading a transient thermal grating

Model assumptions

One Fourier component of a gray two-rate phonon Boltzmann equation: every phonon has the same speed, 10 micrometres per nanosecond, resistive scattering relaxes toward the local temperature and normal scattering toward a drifting distribution that keeps the heat current. Speed and both scattering times are assumed inputs chosen to show the regimes; they are not fitted to graphite or any other crystal. The detector record is the model grating amplitude plus seeded Gaussian noise; a real measurement records surface displacement and reflectivity, which adds terms this model leaves out. The pump crossing angle is drawn ten times larger than the real one.

Grating amplitude after the pump

Pink dots: noisy simulated record. Yellow: solver without noise. Green: exponential fit. Blue dashes: Fourier. Purple dots: free flight.

Fitted diffusivity against grating period

Pink: fit to each record (bars: two standard errors; rings: held out). Blue dashes: Fourier. White line: this period.

Where the phonons are heading, now

Yellow: hot-stripe centre (dashes: local temperature). Orange: between stripes (dashes: drifting equilibrium with the same heat flux).

More numbers and how they are checked

Fourier diffusivity of the model: $0$
Diffusivity from an exponential fit, with one standard error: $0$
Resistive rate < second-sound frequency < normal rate: $0$
Second-sound speed of the model: $0$
Time of the lowest amplitude: $0$
Diffusion: error on held-out periods: $0$
Damped wave: error on held-out periods: $0$
Error a perfect model would show from noise alone: $0$
Damped-wave fit: speed, relaxation time and short-period damping: $0$
Swept periods where the fitted wave swings below zero: $0$
Pump crossing angle for this period: $0$
Length of the record: $0$
Detector noise level: $0$

Five of the seven periods fit each model; the other two are held out. A model earns the explanation only if it predicts the held-out records to within the noise floor. A heavily damped wave is just diffusion, so second sound also needs the fitted wave to swing below zero at the swept periods. The solver itself is checked against a separate Laplace-transform solution in the route tests.

Physics tutorial

Can a temperature pattern ring?

BackgroundIn a transient thermal grating experiment, two short laser pulses cross on a sample and their interference heats it in parallel stripes. A probe beam then reads how fast the stripes fade. Diffusion predicts a smooth exponential decay whose rate grows as the period shrinks.

Why it mattersHuberman and colleagues used this method on graphite near 100 K and saw the signal dip below zero: for a moment the hot stripes were colder than the cold ones. That is second sound, heat moving as a wave, and it needs phonon collisions that conserve momentum.

Start with the essentials

Focus question: Paint a striped temperature pattern with two laser pulses. Does it fade like diffusion, or swing hot and cold like a wave?
One-sentence intuition: Two scattering clocks decide. Resistive scattering (time $\tau_R$ ) destroys the heat current; normal scattering (time $\tau_N$ ) only reshuffles it. A wave appears when the second-sound frequency $\Omega=qv/\sqrt3$ sits inside $1/\tau_R<\Omega<1/\tau_N$ . Free flight alone also makes a small dip, so a dip by itself does not prove second sound.

Core mathematical model

Diffusive decay of one grating

\theta(t)=\theta(0)\,e^{-\alpha q^2 t},\qquad q=\frac{2\pi}{L},\qquad \alpha=\frac{v^2\tau_R}{3}

In this gray model the Fourier diffusivity contains only $\tau_R$ : normal scattering cannot relax the heat current, so it does not change the long-time conductivity.

Second sound when normal scattering is fast

\ddot\theta+\frac{\dot\theta}{\tau_R}+\frac{v^2}{3}\,q^2\,\theta=0

A damped wave with speed $v/\sqrt3$ . It oscillates when $qv\tau_R/\sqrt3>1/2$ .

Common difficulties

A sign flip is not proof by itself

Typical misconceptionAny dip below zero means heat moved as a wave.

Better mental modelWith no scattering at all, phonons simply fly apart and the grating amplitude follows $\sin(qvt)/(qvt)$ , which dips to about $-0.22$ . Second sound gives a deeper, slower swing at speed $v/\sqrt3$ . The held-out check in the data row asks which model predicts periods it never saw, and at how many swept periods the fitted wave really swings below zero.

Run the experiment

01
See the stripes swap
Start with Second sound: stripes swap and watch the stage.
What to observe: Amber and blue stripes trade places and back while the yellow heat-flux arrows reverse; the record dips deep below zero.
02
Compare with free flight
Choose Free flight: a small dip.
What to observe: The record follows the purple free-flight curve, with one shallow dip and fast ringing that dies out; the direction chart shows sharp ripples instead of a straight drifting line.
03
Shrink the period
Choose Short period: decay slows, then Long period: Fourier returns.
What to observe: At 2 µm the fitted diffusivity drops well below the Fourier value; at 40 µm it returns to it. The sweep chart shows the same trend across seven periods.

Transient Thermal Grating: When Heat Rings Like Sound

Can a temperature pattern ring?

Core mathematical model

Diffusive decay of one grating

Second sound when normal scattering is fast

Common difficulties

A sign flip is not proof by itself

Run the experiment

See the stripes swap

Compare with free flight

Shrink the period

Related experiments and articles

From phonon flights to second sound, interfaces and thermoelectrics