B-X4 · Measure energy exchange

Neutron Triple-Axis Spectrometer

What does a neutron’s final energy tell us about motion inside a sample? A neutron that leaves with less energy has excited motion in the sample. Scanning its energy loss reveals excitation peaks, and the energy-gain partner shows how thermal motion populates those excitations.

Select the incident and outgoing neutron energies with two crystals. Scan energy transfer at fixed momentum transfer, record counts and fit an excitation with its thermal partner.

Instrument setup

Measure energy left in the sample

01Scattering setting

Keep momentum transfer fixed as the energy scan turns the instrument.

Fixed momentum-transfer magnitude

Q/\mathring{\mathrm A}^{-1}

2

02Sample excitation

These are the teaching sample’s source inputs. The fit estimates from the stored counts.

Source excitation energy

\varepsilon_0/\mathrm{meV}

3

Source temperature

T/\mathrm K

80

Calibration, resolution and counting

Broader response blends the peaks; more exposure reduces counting noise.

Instrument energy response standard deviation

\sigma_R/\mathrm{meV}

0.35

Incident monitor scale

C

1600

Changing the setup keeps the previous records and marks them as a comparison. The probe beside the 3D bench only changes the view of the instrument.

A nonrelativistic neutron beam scatters from an illustrative thermal harmonic excitation. Final energy is fixed at 14.7 millielectronvolts. Every scan point solves the momentum triangle and the Bragg angles of a schematic 3.355-angstrom monochromator and analyzer. The sample spectrum contains an elastic Gaussian and one pair of excitation lines with a fixed intrinsic standard deviation of 0.18 millielectronvolts. Gaussian energy response, the outgoing-to-incoming wavevector factor, a flat background and Poisson counts are included. The fit uses those counts and known response widths. It searches excitation energies from one to 5.5 millielectronvolts. This is an energy-only response, not a full TAS resolution volume, a material identification or a reproduction of historical measurements. Magnetic scattering, absorption, multiple scattering, phonon eigenvectors and structure refinement are omitted. Hardware lengths are schematic and increase near backscattering to keep the two crystal mounts and open apertures apart. Particle motion is a slowed illustration; the spectra come from the recorded count model.

Turn the energy-transfer probe

\varepsilon/\mathrm{meV}

3

Drag the gold analyzer handle or move the transfer below. Both energy selectors stay connected to the sample.

Neutron energies

\begin{aligned}E_i&=17.7\,\mathrm{meV}\\E_f&=14.7\,\mathrm{meV}\\Q&=2.00\,\mathring{\mathrm A}^{-1}\end{aligned}

Crystal and sample angles

\begin{aligned}\alpha&=41.6^\circ\\\theta_M&=18.7^\circ\\\theta_A&=20.6^\circ\end{aligned}

Arm lengths are schematic; rotation angles follow the scattering geometry.

Apparatus scale and particle motion

Arm directions and crystal planes follow the solved Bragg geometry. Hardware lengths and moving particles are schematic. The spectrum stays in the data row below.

Measurement sequence

Scan the energy. Find the excitation.

One hundred sixty-one energy settings follow the same fixed momentum transfer. Each count keeps its own solved scattering geometry.

01Set the instrument
02Record the scan
03Infer from counts

Ready. Choose a setup, then start the scan.

Start a scan to record counts. The spectrum will appear here as measurements arrive.

Recorded counts

Neutron energy transfer

No acquisition yet

$\varepsilon/\mathrm{meV},\quad N_{\rm bin}$

Click to move the instrument probe. Teal dots: counts. Purple: fit. Gold dashed: separate generating mean.

Test the measurement model

What remains after fitting?

$\varepsilon/\mathrm{meV},\quad N-\widehat N$

Residuals use the same retained counts. The fit assumes a known Gaussian energy response.

Known instrument response

Peak width needs a resolution scale

$\delta E/\mathrm{meV},\quad R/R_{\max}$

Purple: fixed intrinsic profile. Gold: instrument-broadened profile. These are known response shapes, not acquired counts.

\sigma_{\rm int}=0.18\,\mathrm{meV}

The response width and the fixed intrinsic width combine before counts are generated. A broad, overlapping thermal pair can still give a visible peak while leaving temperature unresolved.

The thermal comparison uses integrated peak areas with the wavevector factor removed. No calibration or model uncertainty is included in the local counting errors.

Start with the essentials

Focus question: What does a neutron’s final energy tell us about motion inside a sample?
One-sentence intuition: The energy the neutron loses is the energy the sample gains.

Core mathematical model

Energy and momentum given to the sample

\mathbf Q=\mathbf k_i-\mathbf k_f,\qquad\varepsilon=E_i-E_f,\qquad E=\frac{\hbar^2k^2}{2m_n}

Positive energy transfer excites the sample. Wavevectors use radians per angstrom, so the wavelength is a full turn divided by the wavevector magnitude.

The scattering triangle must close

\cos\alpha=\frac{k_i^2+k_f^2-Q^2}{2k_i k_f},\qquad2d\sin\theta_B=\lambda

The sample scattering angle is the full angle between the two beams. Each crystal uses its own Bragg half-angle to select energy.

Thermal occupation sets the two integrated areas

\frac{A_{\rm gain}}{A_{\rm loss}}=e^{-\varepsilon_0/(k_B T)}

The fitted amplitudes correct the final-to-incident wavevector factor. This relation uses integrated lines, not the two raw peak heights.

Common difficulties

A reconstructed result depends on the measurement model

Typical misconceptionThe width of the measured peak is the sample’s intrinsic width.

Better mental modelThe instrument response also broadens the spectrum. The fit here uses a known Gaussian energy response and a fixed intrinsic width.

Run the experiment

01
Follow both energy selectors
Move the energy-transfer probe and inspect the source and collector views.
What to observe: The monochromator changes incident energy; the analyzer keeps final energy fixed. The sample arm moves to keep momentum transfer fixed.
02
Measure the thermal pair
Acquire and analyze Resolved excitation. Then compare Cool the sample.
What to observe: Cooling suppresses the energy-gain line because fewer excitations are already present.
03
Test the geometry
Choose Unreachable transfer and try to acquire.
What to observe: The scan stops when part of the chosen momentum and energy transfer cannot form a scattering triangle.

Physical conclusion