Quantum evidence · nuclear resonance

Mössbauer Doppler Observatory

Why can a source moving a few millimeters per second resolve tiny shifts in nuclear energy levels? Because nuclei held in a solid can emit and absorb gamma rays without recoil, and these lines are extremely narrow. A few millimeters per second of source motion is enough to Doppler-shift the line across the absorber's resonances. The dips in transmission against velocity then reveal a quadrupole doublet and how far its center is shifted.

Move a gamma source toward and away from an absorber and count the gamma rays that get through at each speed. The transmission spectrum shows a quadrupole doublet; work out its energy shift relative to the source from the raw counts.

Prepare, measure, then question the result

Why can a source moving a few millimeters per second resolve tiny shifts in nuclear energy levels?

Predict the change before you scan; the raw counts are kept afterward.

Absorber shift relative to source

\delta/(\mathrm{mm\,s^{-1}})

0.25

Quadrupole separation

\Delta v/(\mathrm{mm\,s^{-1}})

1

Recoilless source fraction

f_{\mathrm s}

0.8

Resonant optical depth

\eta

0.16

Extra Lorentz FWHM

w_{\mathrm{extra}}/(\mathrm{mm\,s^{-1}})

0

Exposure per reading

\tau/\mathrm{s}

2

Source velocity

v/(\mathrm{mm\,s^{-1}})

0

Moving the probe keeps the data. Changing the source or the instrument starts a new run. Saved runs stay until you reset.

The source emits one ideal line. The absorber is thin and randomly oriented, and it absorbs as a doublet. The model convolves the source and absorber linewidths. It does not simulate magnetic sextets, saturation in thick absorbers, the phonon spectrum, thermal shifts or errors in the velocity calibration.

Schematic geometry with enlarged gaps and parts. The colors and moving markers do not show visible light or measured trajectories. Drag to orbit.

Exposure: signal and reference counted separately

Give the spectrum a record.

Each coordinate has a signal exposure and a separate reference: absorber removed. Exposure time controls Poisson noise; scan playback is accelerated.

Ready. Predict, then acquire.

Recorded points

0

Probe energy change · model

0

Convolved FWHM · model

0

Measured spectrum in counts per second

Which structure survives measurement?

$v/(\mathrm{mm\,s^{-1}})$ · $N_i/\tau_i\;(\mathrm{s^{-1}})$

Teal: source expectation. Gold: acquired rates with fitted Poisson standard deviations, once a fit exists. Purple dashes: inference from records. Gray dots: saved run. Background is included.

Click or drag to move the probe; its slider supports keyboard input. The vertical scale follows the data.

Inference with a joint Poisson likelihood

Estimate from both exposures.

Inferred absorber shift

\text{—}

Inferred source rate

\text{—}

Deviance per degree

\text{—}

Flat alternative

\text{—}

$r_i=(N_i-\widehat\mu_i)/\sqrt{\widehat\mu_i}$

Gold: signal residuals. Blue: reference residuals. Known response, splitting/width controls and background; unknown source rate and spectral parameter. At least sixteen distinct probe positions are required. Descriptive residuals and deviance do not establish identifiability, confidence intervals or discovery significance.

Keep the original counts.

Save a full or partial run, with its settings and seed, for as long as this page stays open.

#	$v/(\mathrm{mm\,s^{-1}})$	Signal	Absorber removed	$\tau/\mathrm{s}$

The latest six points are shown. The CSV has every raw count, exposure, setting and unit, plus the seed. Moving the probe never changes data already taken.

Scientific scope, geometry & primary sources

A moving single-line gamma source illuminates the absorber and then the upstream face of the transmission counter. Positive velocity means the source moves toward the absorber and raises photon energy. The absorber shift is defined relative to this source, with the same sign.

The photon energy, rounded to 14.4 keV, and the 141 ns mean nuclear lifetime set the natural linewidth. The source and absorber have equal Lorentzian widths, which add under convolution. The absorber gives an isotropic quadrupole doublet with lines of equal strength. Extra Lorentzian velocity broadening widens the lines and makes them shallower, but the integrated absorption area stays the same.

Transmission is first order in a resonant optical depth no larger than 0.2. Before convolution the omitted exponential term is bounded by two percentage points. The source recoilless fraction is adjustable; absorber recoilless weight is included in optical depth. Nonresonant source photons form a flat background. No six-line magnetic pattern, full lattice dynamics, temperature shift or dead time is included. The reference exposure removes the absorber.

Independent Poisson exposures, fixed calibrated background and stable source. No drift, dead time or calibration uncertainty. The source follows the displayed model; successful recovery validates the simulator, not the historical discovery.

MIT · Mössbauer laboratory UCSB · recoilless resonance Maryland · experimental scope

Start with the essentials

Focus question: Why can a source moving a few millimeters per second resolve tiny shifts in nuclear energy levels?
One-sentence intuition: Fit the signal counts and the reference counts together with one Poisson likelihood, and estimate the source rate from the same data.

Core mathematical model

Source velocity as an energy axis

\Delta E=E_\gamma\frac{v}{c},\qquad \Gamma_E=\frac{\hbar}{\tau},\qquad \Gamma_v=\frac{c\Gamma_E}{E_\gamma}

A positive source velocity means the source moves toward the absorber. The lifetime here is the mean lifetime, not the half-life.

Transmission through a thin absorber

W=2\Gamma_v+w_{\mathrm{extra}},\qquad g(v)=1-\frac{f_{\mathrm s}\eta\Gamma_v}{2W}\sum_{q=\pm1}\frac{1}{1+4(v-\delta-q\Delta v/2)^2/W^2}

This thin-absorber result combines the source and absorber linewidths, taken as equal. Extra Lorentz broadening keeps the absorption area the same, so the dips get shallower.

Recoil energy of a free nucleus

E_R\simeq\frac{E_\gamma^2}{2Mc^2}

A free nucleus recoils. In recoilless emission or absorption, the lattice takes up the momentum and the recoil energy is negligible. Creating phonons is a separate process.

Two independent exposures

N_i\sim\operatorname{Poisson}\!\left[\tau_i(Sg_i+b)\right],\qquad M_i\sim\operatorname{Poisson}\!\left[\tau_i(S+b)\right]

The background rate is known in advance. The source rate and the spectral parameter are fitted together, and the scatter in the reference counts is included. Counts are drawn directly from the Poisson distribution, not rounded from a Gaussian approximation.

Common difficulties

A good fit does not test the law

Typical misconceptionIf the fit matches the data, that independently confirms the physical law built into the simulator.

Better mental modelThe simulator generates the data from a response it is given. Recovering the parameter shows that this simulated measurement and analysis work together. The deviance and residuals describe how well the curve fits; they are not confidence intervals, significance tests or an outside scientific certification.

Run the experiment

01
Scan both sides
Scan the doublet and save the run. Compare the fitted center with the absorber shift you set.
What to observe: The center is measured relative to the chosen source. The detector counts gamma rays that pass through the absorber, so a resonance lowers the count.
02
Separate width from depth
Choose the Broadened drive preset. Scan and compare with the saved doublet.
What to observe: Convolving with an extra normalized Lorentzian makes the dips wider and shallower. The absorption area is fixed, so the depth cannot stay the same.
03
Remove the recoilless source
Set Recoilless source fraction to zero and scan. Then restore the saved run.
What to observe: The spectrum goes flat and no shift is estimated. Momentum is still conserved in this case.

Mössbauer Doppler Observatory

Give the spectrum a record.

Which structure survives measurement?

Estimate from both exposures.

Keep the original counts.