---
title: "Reactor Near–Far Spectra Interactive Lab | Sandbox Physics"
description: "How can detectors near and far from a reactor reveal neutrinos disappearing on the way? Compare the far detector's spectrum with the near one's."
canonical_url: "https://sandboxphysics.com/labs/reactor-near-far-spectra/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/reactor-near-far-spectra/"
last_modified: "2026-10-27"
---

Neutrinos · D06 · A deficit in the spectrum

# Reactor Near–Far Spectra

**How can detectors near and far from a reactor reveal neutrinos disappearing on the way?** Compare the far detector's spectrum with the near one's. A dip at the far site can mean neutrinos disappeared on the way, but a slightly wrong energy calibration, or an error in scaling one detector's rate against the other's, can fake a similar deficit. The near detector pins down the flux both detectors share; it does not cancel every difference in response or background.

Record prompt-energy spectra at a near and a far detector, then fit the disappearance amplitude. Compare the effects of energy resolution, exposure and calibration, and refit the same recorded counts under different assumptions.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

This teaching simulation has one point-like reactor, one target at the near site and four equal targets at the far site. A smooth illustrative source spectrum, the leading-order energy relation for inverse beta decay, a Gaussian response, a fixed mass splitting and a known background replace real reactor and detector models. A joint Poisson fit adjusts a common flux and a constrained relative normalization. The intervals are deviance contours that hold only under these assumptions; their coverage is not calibrated and no discovery significance is given. The Lab does not reproduce Daya Bay, RENO, Double Chooz or JUNO data.

Inference from fixed counts depends on assumptions

## The dip is small, so the comparison must be careful.

Near counts Far counts Fitted means Refitted no-disappearance reference Reveal generating means

### Near: constrain the common flux

Dots are sampled counts. Vertical bars show the square root of the count as a descriptive scale, not exact Poisson intervals. Lines include the known background.

### Far: resolve the missing fraction

Dots are sampled counts. Vertical bars show the square root of the count as a descriptive scale, not exact Poisson intervals. Lines include the known background.

### Compare shape after distance and target correction

Divide the background-subtracted far counts by the near counts and the geometric target factor. Dashed: the assumed unoscillated response ratio before a normalization pull. Delta-method bars are approximate; bins with nonpositive near denominators are omitted. The vertical axis is clipped to the displayed range.

### The fit is a landscape

At each amplitude, refit common flux and far relative normalization. The minimum is gold; shading marks a deviance rise of one. Values above twelve are clipped. Pink appears only when generating means are revealed.

### Inspect one energy bin

$0$

- **Near counts**: $0$
- **Far counts**: $0$

Select a spectrum bin with a click or arrow keys. The record remains unchanged.

### Read the conditional result

- **Deviance-rise-one interval**: $0$
- **Common flux multiplier**: $0$
- **Far relative normalization**: $0$
- **Minimum joint deviance**: $0$

Bin counts and fitted means as a table

Prompt-energy bins in MeV, including response tails

| Energy | Near | Far | Near fit | Far fit |
| --- | --- | --- | --- | --- |

Conventions, model boundaries & primary sources

One point reactor feeds both stations. Four equal far target modules compensate partly for inverse-square dilution. Scene coordinates, cavern sizes and routes are schematic; baseline numbers control the model. Phototubes and liquid illustrate apparatus, without simulated optical hits or neutron transport. All four far modules use one nominal baseline.

The positive fixed splitting is 0.00245 eV squared. The leading short-baseline disappearance amplitude is scanned from zero to 0.30; solar oscillations, matter, isotope evolution, source-shape errors and nonlinear response are omitted. A smooth illustrative source times leading-order IBD phase space is integrated from threshold to 10 MeV. Gaussian prompt-energy tails are retained in two overflow bins and included in the likelihood. The model sets 9,000 unoscillated near counts per exposure at a half-kilometer baseline.

Poisson sampling occurs after response convolution. The background is a known flat mean, not a measured subtraction. Both spectra constrain one freely fitted common flux. A Gaussian penalty applies to the logarithm of the far relative normalization; its width is the control value. Calibration is fixed to the analysis setting, not profiled. The interval conditions on all other model assumptions and is not an experimentally validated confidence statement.

Injected amplitude：$0$

- [Vogel & Beacom · Leading-order inverse beta decay](https://arxiv.org/abs/hep-ph/9903554)
- [Daya Bay · 2012 original disappearance paper](https://arxiv.org/abs/1203.1669)
- [PDG 2024 · §14.4 / §14.6.4](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf)

[Read the near–far story →](https://sandboxphysics.com/stories/neutrinos/daya-bay-near-far/) [How did a candidate become a count? →](https://sandboxphysics.com/labs/reactor-coincidence/)

Physics tutorial

## Measure a deficit by making a comparison

**Background** A reactor emits electron antineutrinos. Over kilometer distances a small fraction disappears, by an amount that depends on energy, and this changes the prompt spectrum. The near detector helps pin down the source normalization that both detectors share.

**Why it matters** Daya Bay, RENO and Double Chooz compared detectors at different distances and kept the differences between detectors under control. This Lab shows the spectral comparison on its own, with a deliberately simpler source, geometry and response.

Start with the essentials

- **Focus question**: Could the same deficit at the far detector come from disappearance, from calibration or from the relative normalization?
- **One-sentence intuition**: A near detector pins down the flux that both detectors share. It does not cancel every difference in response, every background uncertainty or every error in the source shape.

### Core mathematical model

#### Disappearance at short baselines, to leading order

$$
P_{ee}(E,L)=1-A\sin^2\!\left(1.267\,\frac{\Delta m^2}{\mathrm{eV}^2}\frac{L/\mathrm{m}}{E/\mathrm{MeV}}\right),\qquad A\simeq\sin^2 2\theta_{13}
$$

The mass splitting is fixed at a positive 0.00245 eV squared. Baselines are entered in kilometers and converted to meters. Solar and matter corrections are left out, so this is a leading-order approximation, not a full three-flavor fit.

#### Expected counts in each energy bin

$$
\mu_{di}=\alpha f_d\,\frac{C_d T}{L_d^2}\int \Phi(E)\sigma(E)P_{ee}(E,L_d)R_{di}(E)\,dE+B_i
$$

Integrating a Gaussian response over each bin gives the response probabilities, with underflow and overflow included. The far station has four times the target of the near station. The common flux multiplier is free, while the logarithm of the far relative multiplier is constrained. The source and response are illustrative.

#### Fit statistic for the sampled counts

$$
D=2\sum_{d,i}\left[\mu_{di}-n_{di}+n_{di}\ln\frac{n_{di}}{\mu_{di}}\right]+\frac{\eta^2}{\sigma_\eta^2},\qquad f_n=1,\quad f_f=e^\eta
$$

For a bin with zero counts, the log term is taken as zero. At each scanned amplitude, the common flux and the relative log-normalization are fitted again. All other parameters, including the calibration and the known background, stay at their assumed values.

#### Interval from a deviance rise of one

$$
D_{\mathrm{prof}}(A)-D_{\mathrm{min}}=1
$$

The shaded interval uses this descriptive threshold. The amplitude scan has physical and numerical boundaries, and neither its coverage at finite counts nor its behavior near those boundaries has been calibrated. No discovery significance is drawn from it.

### Common difficulties

#### The near detector also sees disappearance

**Typical misconception** The near spectrum is simply the original reactor spectrum.

**Better mental model** The near station is still some distance from the reactor, so the model includes disappearance there too. It also has its own response and background.

#### A fit cannot make up for effects left out

**Typical misconception** A narrow interval proves the model is right.

**Better mental model** The contour assumes a given source shape, a known background and a fixed calibration. Effects outside the model can bias even a precise result.

### Run the experiment

1. 01

   #### Record both spectra

   Start with Near–far. Look at both spectra and the corrected ratio.

   **What to observe:** The near and far counts differ for three reasons: the flux thins out with distance, the target masses differ and some antineutrinos disappear. Reading the ratio requires understanding all three.

2. 02

   #### Increase the exposure

   Choose More counts and compare the width of the profile.

   **What to observe:** More events usually tighten the statistical limit. The uncertainty in the relative normalization still limits the comparison.

3. 03

   #### Change an assumption, not the data

   Increase Relative normalization uncertainty without acquiring again.

   **What to observe:** The normalization can now absorb more of the deficit, so the interval changes on the same data.

4. 04

   #### Try the calibration trap

   Choose Calibration trap, then set Assumed far energy-scale offset to two percent.

   **What to observe:** Matching the injected calibration changes the fit, though no sampled count changes. The deliberately wrong model can give a biased result.

5. 05

   #### Compare the truth with the fit

   Reveal the generating means, then export the counts and the profile scan.

   **What to observe:** The simulation truth is shown only as a separate reference. The fitted curves come from the sampled spectra; the fit never reads the injected amplitude.

Physical conclusion

The evidence comes from a careful comparison of the two sampled spectra, with the statistical scatter and the response assumptions kept in view.

## Related experiments and articles

Continue exploring

### How flavor change is predicted and tested

Propagation through vacuum and matter predicts a flavor probability. How do solar, atmospheric, reactor and beam experiments test it?

1. [Lab **Neutrino Flavor Propagation** Start with coherent vacuum propagation and ideal flavor trials.](https://sandboxphysics.com/labs/neutrino-flavor-propagation/)
2. [Lab **Neutrino Matter Resonance** Follow a rotating matter Hamiltonian and compare slow with sudden conversion.](https://sandboxphysics.com/labs/neutrino-matter-resonance/)
3. [Lab **Solar Neutrino Flavor Observatory** Separate electron flavor from total active solar flux with three reaction responses.](https://sandboxphysics.com/labs/solar-neutrino-flux/)
4. [Lab **Atmospheric Neutrino Zenith Experiment** Turn Earth-crossing baselines into direction-dependent event counts.](https://sandboxphysics.com/labs/atmospheric-neutrino-zenith/)
5. [You are here **Reactor Near–Far Spectra** Fit the same near and far counts while testing calibration and normalization.](https://sandboxphysics.com/labs/reactor-near-far-spectra/)
6. [Lab **Long-Baseline CP Observatory** Compare both accelerator beam polarities on frozen spectra and separate CP phase, matter and response assumptions.](https://sandboxphysics.com/labs/long-baseline-cp/)

Neutrinos · Reading path

### Read reactors and the Earth

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Daya Bay: why compare a near detector with a far one?**](https://sandboxphysics.com/stories/neutrinos/daya-bay-near-far/)
2. [Lab **Reactor Antineutrino Coincidence**](https://sandboxphysics.com/labs/reactor-coincidence/)
3. [Article **RENO and Double Chooz: why repeat the mixing-angle measurement?**](https://sandboxphysics.com/stories/neutrinos/reno-double-chooz-independent-mixing-tests/)
4. [Lab **Neutrino Flavor Propagation**](https://sandboxphysics.com/labs/neutrino-flavor-propagation/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)