Coherent Elastic Neutrino–Nucleus Scattering Cross Section Calculator

Introduction: CEvNS explanation and cross-section estimate

Coherent elastic neutrino–nucleus scattering (CEvNS) is a low-energy neutral-current process in which a neutrino transfers momentum to an entire nucleus instead of to a single nucleon. When the neutrino wavelength is long compared with the nuclear size, the scattering amplitudes from the protons and neutrons add coherently, and the cross section grows with the square of the nucleus's weak charge. That coherence is what makes CEvNS unusually important in reactor monitoring, supernova studies, and precision tests of the weak interaction. This calculator turns the neutrino energy and target composition into a leading-order CEvNS cross-section estimate using the Standard Model approximation implemented below.

The governing formula comes from electroweak theory. For a nucleus with Z protons and N neutrons, the weak charge is , where sin²θ_W≈0.231. The differential cross section for a neutrino of energy E scattering from a heavy nucleus of mass M into recoil energy T is given by . For recoil energies small compared with the neutrino energy, integration over T from 0 to T_max yields a total cross section proportional to GF² QW² M Tmax / 4π. The final result is converted from natural units (GeV⁻²) to square centimeters for convenience. Although the cross section is tiny, typically 10⁻³⁹ to 10⁻³⁷ cm², it is orders of magnitude larger than neutrino–electron scattering in the same energy range, explaining why CEvNS was observed only recently despite decades of experimental effort.

Understanding the scaling with Z and N highlights the physics. Since the weak charge is dominated by the neutron number, heavy neutron-rich nuclei such as cesium or xenon provide the largest signals. The near cancellation of the proton contribution due to 1−4sin²θ_W ≈ 0.076 means that even doubling Z increases the cross section only modestly, whereas adding neutrons squares the amplitude. This is why CEvNS detectors often employ materials with high mass numbers. The dependence on E² arises because higher energy neutrinos can transfer more momentum while still maintaining coherence. However, when the momentum transfer exceeds roughly 1/R, where R is the nuclear radius, the coherence is lost and the form factor suppression becomes significant, reducing the cross section.

The phenomenon has practical implications far beyond the Standard Model curiosity. In core-collapse supernovae, CEvNS dominates the opacity of neutrinos in the dense stellar core, influencing the dynamics of the explosion and the emergent neutrino spectra. In the realm of particle physics, precise measurements of CEvNS can test the weak mixing angle at low momentum transfer and search for non-standard interactions or sterile neutrinos. The process also underlies neutrino-driven neutronization of heavy elements and contributes to the cooling of neutron stars through neutrino emission. Security applications exploit the relatively large cross section to design compact neutrino detectors capable of monitoring nuclear reactors from a distance, providing a non-intrusive safeguard mechanism.

To anchor the calculations, consider the default parameters representing a germanium nucleus (Z=32, N=40). For a 30 MeV neutrino—a typical energy in a stopped-pion source—the calculator yields a cross section of order 10⁻³⁸ cm². If we change to a lighter nucleus such as carbon (Z=6, N=6), the weak charge drops dramatically and the cross section shrinks by nearly two orders of magnitude. Conversely, switching to xenon (Z=54, N=77) pushes the cross section to ~10⁻³⁷ cm², explaining why large liquid xenon detectors can observe CEvNS signals despite the low fluxes. The dependence on E² means that supernova neutrinos with energies around 10 MeV scatter an order of magnitude less efficiently than spallation-source neutrinos, but the immense flux during a galactic supernova compensates, making CEvNS a key channel for neutrino detection in future observatories.

A representative table summarizing cross sections for different nuclei at 30 MeV illustrates these trends.

These orders of magnitude match those reported by experimental collaborations such as COHERENT, which achieved the first observation of CEvNS in 2017 using a cesium iodide detector. The measured cross section agreed with Standard Model predictions within uncertainties, confirming the coherence enhancement. Subsequent experiments with argon, germanium, and liquid xenon targets continue to refine the measurements, aiming to constrain new physics. Any deviation from the predicted dependence on N and E could hint at exotic interactions or the presence of light mediators coupling to neutrinos.

The derivation of the CEvNS cross section relies on a few key approximations. First, it assumes the nucleus remains in its ground state; inelastic processes such as excitation or breakup are neglected. This assumption holds when the recoil energy is much smaller than nuclear excitation energies, which is typically the case for neutrino energies below ~50 MeV. Second, the form factor is taken to be unity. In reality, the distribution of nucleons within the nucleus reduces coherence at higher momentum transfer. The inclusion of realistic form factors can lower the cross section by tens of percent for heavier nuclei or higher energies. Third, radiative corrections and quenching effects are ignored, though they can be incorporated for precision work. The calculator provides a baseline estimate suitable for initial feasibility studies and pedagogy.

From a theoretical standpoint, CEvNS offers a rare window into the weak neutral current at low energies. Unlike charged-current processes, it does not rely on the details of nuclear structure and therefore provides a clean probe of fundamental couplings. The process is also sensitive to the neutron distribution within nuclei, linking it to studies of the neutron skin and equation of state of neutron-rich matter. By measuring the recoil spectrum with high precision, experiments can extract the weak form factor and hence information about the spatial arrangement of neutrons, complementing parity-violating electron scattering techniques.

Looking ahead, CEvNS could play a role in astrophysical neutrino detection. Coherent scattering off the Earth or the Moon could produce tiny recoil signals detectable by future seismometers, potentially enabling geoneutrino studies. In the context of dark matter searches, CEvNS from solar or atmospheric neutrinos constitutes an irreducible background known as the “neutrino floor,” setting a limit to the sensitivity of direct detection experiments. Accurately modeling the CEvNS cross section is therefore essential for interpreting potential dark matter signals.

In summary, coherent elastic neutrino–nucleus scattering exemplifies how subtle quantum mechanical effects can amplify weak interactions into measurable phenomena. The quadratic scaling with neutron number and the coherent enhancement open avenues for compact neutrino detectors and precision tests of the Standard Model. The calculator here provides a convenient way to explore how different target materials and neutrino energies influence the cross section, serving as a springboard for deeper investigations into neutrino physics and its interdisciplinary applications.

Recording Your CEvNS Estimate

Use the copy button to save the weak charge and total cross section for the current CEvNS target. Keeping those values together makes it easier to compare nuclei, track the effect of changing the neutrino energy, or paste the result into detector notes and analysis tables.

How to use this CEvNS calculator

1. Enter Neutrino energy (MeV) in MeV, matching the low-energy neutrino range the CEvNS approximation expects.
2. Enter Atomic number Z as the proton count of the nucleus you want to model.
3. Enter Neutron number N as the neutron count for that same target nucleus.
4. Run one CEvNS estimate, then change only a single input before running it again so the comparison stays meaningful.

Formula: the CEvNS estimate used by this calculator

The result is built from the leading-order coherent elastic neutrino–nucleus scattering expression implemented in the script: the weak charge is Q_W = N - (1 - 4 sin²θ_W)Z, the maximum recoil is T_max = 2E² / (M + 2E), and the total cross section is approximated as σ ≈ G_F² Q_W² M T_max / 4π before converting from GeV⁻² to cm². The mass input A is treated as an approximate nuclear mass in amu, so the calculator stays lightweight and gives a baseline CEvNS estimate rather than a full nuclear-structure treatment.

Worked example: comparing a light and heavy CEvNS target

To see the coherence effect clearly, hold the neutrino energy fixed and compare a light nucleus with a heavy neutron-rich target. The heavier nucleus benefits much more from the Q_W² term, so its total cross section rises sharply even when the neutrino energy stays the same. If you change only Z while N remains modest, the result moves less, because the proton contribution is partly cancelled by 1 − 4 sin²θ_W. That is the main reason CEvNS favors neutron-rich materials when you are trying to maximize the signal.

Limitations and assumptions for CEvNS estimates

This CEvNS calculator is a leading-order estimate, not a complete detector or event-rate simulation. It assumes the nucleus remains coherent, uses a fixed sin²θ_W value, takes the nuclear form factor to be unity, and ignores inelastic channels, radiative corrections, quenching, and detector thresholds. The answer is only as reliable as the target inputs you supply: if E, Z, N, and A do not describe the same nucleus and energy regime, the reported cross section can be misleading. For precision work, compare the result with experiment-specific nuclear data, a structure calculation, or a dedicated Monte Carlo model before drawing conclusions.