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A search for interactions of passing dark matter with a terrestrial target, for example nuclear recoils from elastic scattering. Predicting a count requires the local density and velocity distribution, interaction cross section, target response and detector acceptance. A background-compatible count gives a conditional interaction bound, rather than a measurement of the cosmological relic density.

High-energy elastic electron scattering probes nuclear charge through the electromagnetic interaction. Resolve the elastic peak with a spectrometer, measure its angular dependence and normalize by beam charge, target thickness and detector acceptance. In a plane-wave, approximately static treatment,Here $q$ is physical momentum transfer divided by $\hslash$, not the momentum itself. The low-$q$ slope, $F(q)=1-q^2⟨r^2⟩/6+\cdots$, determines the charge r.m.s. radius. A sharp uniform sphere hasIts first diffraction minimum is at $qR\simeq 4.493$, illustrating how angular structure determines size. Radii approximately follow $R\propto A^{1/3}$, but a sharp radius, a surface half-density radius and an r.m.s. radius are different definitions.

A motivating experimental implementation is Hofstadter's magnetic-spectrometer measurement: selecting outgoing electron momentum separates elastic scattering from nuclear excitation, rather than treating every scattered electron as an elastic event. Beam-energy and angle calibration, empty-target or blocked-acceptance backgrounds, and target-thickness checks matter. Finite resolution mixes nearby excitations; multiple scattering and Coulomb distortion require correction before a precision density is inferred.^[ref]

Shape information comes from quadrupole moments, Coulomb-excitation transition strengths and rotational spectra. An intrinsic axial moment $Q_0=\int \rho (3z^2-r^2)d^{3}r$ is positive for a prolate charge distribution and negative for an oblate one. A laboratory quadrupole moment is not identical to $Q_0$: a rotationally averaged spin-zero state can have zero laboratory moment despite intrinsic deformation. Collective rotational levels approximately follow $E_J={\hslash }^{2}J(J+1)/(2\mathcal{I})$, while measured $E2$ strengths test the associated charge collectivity. Muonic-atom X-rays and isotope shifts supply complementary charge-radius constraints; hadronic probes provide additional, more interaction-dependent matter and neutron information. Neither a charge radius alone nor a finite-range form-factor measurement uniquely fixes the entire nuclear shape.

supersymmetry predicts partners with spins differing by one half. At a proton collider, strongly interacting squarks and gluinos can be pair produced if kinematically accessible. With conserved R-parity, their cascade decays end in a stable lightest supersymmetric particle, often a neutralino. The generic signature is energetic particle jets, possibly leptons or heavy-flavour jets, together with missing transverse momentum from invisible particles. Longitudinal missing momentum is not an equally clean observable because the initial parton momentum fractions are unknown.

A practical inclusive search defines collider signal regions using jet multiplicity, transverse momenta, missing transverse momentum and an event scale such asLeptonic and heavy-flavour selections probe other decay chains. For a compressed supersymmetric spectrum, the daughter particles can be soft, making a hard initial-state radiation jet useful for triggering and recoil. Direct chargino or neutralino production instead motivates multilepton searches. Decays with R-parity violation can yield high jet or lepton multiplicities with little missing momentum, and long-lived particles motivate displaced vertices, disappearing tracks or anomalous ionization. No single selection covers all these cases.

The difficult issue is background inference. Invisible $Z$ decay with jets is a real missing-momentum background: a photon-plus-jets sample or reconstructed $Z$ decays to charged leptons constrains its normalization and recoil shape through modeled transfer factors. A single-lepton collider control region constrains $W$ and top backgrounds whose charged lepton is missed in the collider signal region. Jet mismeasurement in ordinary multijet events gives instrumental missing momentum; angular cuts, detector-quality checks and control samples test that tail. Validation regions check these extrapolations without tuning on the final signal count.

detector acceptance, trigger efficiency, jet calibration, collider luminosity, theoretical production rates and transfer factors enter a likelihood as nuisance parameters. One compares the observed signal-region distribution with background-only and signal-plus-background hypotheses, using a profile likelihood and reporting exclusions for specified masses, branching fractions and spectra. As a pre-examination example, the 2014 ATLAS zero-lepton jets-plus-missing-momentum search used $20.3\mathrm{f}{\mathrm{b}}^{-1}$ at $8\mathrm{TeV}$ and found no significant excess. Its interpretation depended on R-parity conservation and a neutralino lightest state; different simplified models gave different mass exclusions.^[ref]

A null search excludes the tested parameter region, not supersymmetry as a whole. Conversely, a missing-momentum excess alone would not establish supersymmetry: one would need consistent masses, decay chains, couplings and eventually spin information. A stable particle on detector flight-time scales is not thereby proved stable for the age of the universe.