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.

Where did the neutron’s energy go?

Energy transfer

\varepsilon/\mathrm{meV}

3

Momentum transfer

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

2

Sample temperature

T/\mathrm K

80

Cold samples lose their energy-gain peak.

Simulated preview

Calibration and counting

Source excitation energy

\varepsilon_0/\mathrm{meV}

3

Instrument energy response standard deviation

\sigma_R/\mathrm{meV}

0.35

Incident monitor scale

C

1600

One harmonic excitation, known Gaussian response and Poisson counts. Excitation and temperature are source inputs, excluded from inference.

Model assumptions

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.

Schematic arm lengths and displacement density; transport slowed.

Geometry and visual scales

E_f=14.7\,\mathrm{meV}

\alpha=41.6^\circ

The sample cloud shows the stationary RMS displacement of the assumed thermal oscillator; shell shapes are schematic. Its RMS size scales with Bose occupation, with an arbitrary zero-point length. It is not an atomic trajectory. Count feedback uses the stored detector bin; all arm and crystal angles use the same solved geometry.

Capture a new scan or choose a demo.

The energy exchanged with the sample

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

Teal: counts. Purple: fit. Gold dashed: generating mean. Pink cursor: probe.

Inspect the fit and calibration

Counting residuals

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

Known resolution

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

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

Purple: intrinsic profile. Gold: instrument-broadened response, not counts. Counting errors assume this known response and exclude calibration and model uncertainty.

From arrival times to energy exchange

Saved measurement

Recorded tags · sample, empty and dark

Separate compatible branches; unbounded branches in the table.

Finite spectrum bounds · negative corrected intervals retained

How to read these data

Sample, empty container and electronics dark are separate exposures. Each compatible source origin and disconnected branch remains separate; a plotted region is not a measured point. Counting uncertainty and shared references retain their distinct roles. Negative corrected intervals remain.

Candidates for each actual event

Raw tags remain even when physical assignment fails. A union of source origins is retained as a union.

Tag	Pixel	Raw clock code	Possible source	Energy transfer / meV	Momentum-transfer magnitude / inverse angstrom	Support

Independent clock, geometry and incident references

The selected wavelength and emission time share slit constraints. Reference errors are common to many events and do not disappear by averaging more hits.

Forward replay, source limits and unresolved slow flights

Separate simulated reference · actual curved flight segments and terminal contacts. The inverse does not consume these hidden paths.

A 1.3 second observed beam-off history bounds earlier arrivals under the supplied finite chamber. A 120 millisecond observation still need not drain all allowed slow final flights.

Save or reopen the complete archive

Import

Up to eight immutable measurements, with a cumulative decoded limit of 64 MiB. Raw events, independent references and units survive both formats; imports are checked before the session changes.

Open an empty session

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

Read the instrument principle, sources and limits

Focus question: How can one detector timestamp distinguish the incoming and outgoing flights?
One-sentence intuition: An independent bound on the incident flight lets the remaining time constrain the outgoing flight. Finite slots and pixels produce correlated intervals.

Core mathematical model

Upward vertical coordinates and downward gravity

\mathbf r(t)=\mathbf r_0+\mathbf v_0t+\tfrac12\mathbf gt^2,\qquad\mathbf g=(0,0,-9.80665)\,\mathrm{m\,s^{-2}}

The right-handed axes follow the incident beam, horizontal transverse direction and vertical upward. Each post-sample flight begins with the actual new velocity.

Speed sets kinetic energy

E=\frac12m_nv^2,\qquad\lambda=\frac{h}{m_nv}

Neutron mass and Planck’s constant connect speed with energy and wavelength. The actual trajectory calculation also retains gravity.

Energy and momentum given to the sample

W=E_i-E_f,\qquad\mathbf Q=\frac{m_n}{\hbar}(\mathbf v_i-\mathbf v_f)

Positive energy transfer excites the sample. Both velocities refer to the sample; detector angle alone does not replace the vector relation.

Common difficulties

A late neutron can come from an earlier pulse

Typical misconceptionThe most recent source trigger identifies every detector event.

Better mental modelA slow neutron or a neutron that lost substantial energy can arrive after a later pulse. Every allowed origin must be considered.

Narrower slots also remove counts

Typical misconceptionA narrower opening always makes every spectral feature clearer.

Better mental modelIt narrows the joint speed/release-time range but admits fewer neutrons. Finite pixels, geometry and reference uncertainty still contribute.

Negative background-corrected results

Typical misconceptionA negative corrected count must be a software error.

Better mental modelSample, container and dark observations each have finite counts. Their difference can be negative and retains its counting uncertainty.