Coherent Elastic Neutrino–Nucleus Scattering Calculator

Explanation

Coherent elastic neutrino–nucleus scattering (CEvNS) is a subtle process predicted by the Standard Model in which a low-energy neutrino scatters off an entire nucleus via the weak neutral current. Because the de Broglie wavelength of a neutrino with tens of MeV energy exceeds the nuclear radius, the neutrino sees the nucleus as a single coherent object. The amplitudes from scattering off individual nucleons add constructively, leading to a cross section that scales with the square of the total weak charge. This coherence dramatically enhances the cross section compared to neutrino–electron scattering and plays a vital role in supernova physics, nuclear security applications, and precision tests of the weak interaction. The calculator above provides an estimate of the total CEvNS cross section for given neutrino energy and nuclear composition, following the leading-order expressions used in experimental analyses.

The governing formula originates from the electroweak theory. For a nucleus with Z protons and N neutrons, the weak charge is $\mathrm{Q\_W}=N-(1-4{\mathrm{\backslash sin}}^2\mathrm{\backslash theta\_W})Z$, where sin²θ_W≈0.231. The total cross section for a neutrino of energy E scattering off a heavy nucleus of mass M through an angle that results in recoil energy T is given by $\frac{d\sigma }{dT}=\frac{{\mathrm{G\_F}}^{2}M}{4\pi }{\mathrm{Q\_W}}^2(1-\frac{MT}{2E^2})$. For recoil energies small compared with the neutrino energy—a valid approximation in the coherent regime—the nuclear form factor is close to unity and integration over T from 0 to T_max yields the expression coded above: $\sigma \approx \frac{{\mathrm{G\_F}}^2{\mathrm{Q\_W}}^{2}MT\mathrm{\_\{max\}}}{4\pi }$. 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.