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Calorimetry measures heat transfers. A calorimeter, a calibrated heat capacity and an accounting of unwanted heat transfer connect an observed temperature change to an energy balance, as in heat-engine calorimetry and heat-pump calorimetry.

The central idea is to measure a response that is easier to observe than the transfer itself: a known heat capacity turns a temperature change into an energy estimate. Mixing calorimetry can determine an unknown specific heat capacity, while a chemical reaction can be studied by measuring the response of its surroundings. The useful observable is a change in temperature, together with a stated thermodynamic system boundary; a hot object does not contain a separately conserved store of heat.

For a closed system with unchanged bulk kinetic energy and potential energy, let $Q_{\mathrm{in}}$ be net inward heat and $W_{\mathrm{on}}$ net inward work. The first law of thermodynamics givesIf the boundary includes a hot sample, receiving water and the participating vessel, their mutual heat transfer is internal. With no external input, the total storage change is minus the signed outward heat leakage, not the positive amount transferred from sample to water. A calorimeter calibration with an immersed resistor instead supplies electrical work across this boundary; Joule heating converts it into internal energy. Choosing only the receiving water as the boundary changes which transfers count as external.

The mechanical constraint decides which storage quantity the readout represents. For a closed system at fixed volume, with no other work, $Q_V=\Delta U$. At constant external pressure, with only expansion work $p\Delta V$, $Q_p=\Delta H$, where $H=U+pV$ is enthalpy. Thus the corresponding warming balances use heat capacity at constant volume or heat capacity at constant pressure. Electrical or shaft work must be included separately when present. These endpoint balances do not require the actual exchange to be a reversible thermodynamic process.

A covered cup open to atmospheric pressure approximately realizes the second constraint when escaping material is negligible. A sealed rigid reaction vessel realizes the first; its surrounding water, vessel and immersed parts form the calibrated receiver. The reaction's outward heat equals the receiver's gain only after losses and ignition input have been accounted for. The distinction concerns the reaction boundary, even if its receiver is open to atmospheric pressure.^[ref]

A historical quantitative result appears in James Prescott Joule's On the Mechanical Equivalent of Heat (1850): Table IX gives $772.692$ foot-pounds of mechanical work per warming of one pound of water by one degree Fahrenheit, with weighing reduced to vacuum. His final rounded statement specifies water between $55$ and $60$ degrees Fahrenheit. He acknowledged residual vibration and sound as possible upward bias. This is an original reported conversion result, not the definition of the modern joule; the apparatus and resting controls are described in Joule's paddle-wheel experiment.^[ref]

Additional collisions overlapping an event in the same beam crossing or detector response window. Extra tracks and calorimeter deposits affect particle-jet reconstruction, isolation and missing transverse momentum. Vertex association, subtraction and response calibration reduce the effect but leave uncertainties that depend on collision conditions.

An electrical calorimetry experiment asks whether a measured electrical work input accounts for a measured warming. The University of Iowa's demonstration uses an insulated flask containing water, an immersed heater, a thermometer, electrical meters and a timer. Its procedure compares the duration, electrical readings and temperature rise. The following low-power measurement design and synthetic readout develop that idea; they are not measurements reported by the demonstration. ^[ref]

Use a covered calorimeter with weighed water, an immersed resistor, a calibrated thermometer and gentle stirring. Put an ammeter in series with the heater branch and a high-input-electrical resistance voltmeter across the immersed resistor's terminals. Record $V(t)$, $I(t)$, temperature $T(t)$ and room temperature $T_a(t)$ before, during and after switching the heater on. Measure the electric potential difference at the heater, rather than at the supply: lead and contact electrical resistance outside the receiving boundary can consume power that never reaches the water. Account for voltmeter loading if the ammeter also includes the sensing branch's electric current.^[ref]

The electrical readout gives operating electrical resistance $R(t)=V(t)/I(t)$ when $I(t)\ne 0$, and power $P(t)=V(t)I(t)$. To test Ohm's law, obtain several pairs at the same temperature; one ratio alone cannot establish a linear response. For small electrical resistance, use four-terminal sensing to exclude lead drops and reverse electric current to test stable offsets from the thermoelectric effect. Use brief low-power resistance measurements if the intended result is the unheated value: the test itself causes Joule heating.

For a covered assembly at approximately constant pressure, with negligible evaporation, negligible endpoint mechanical energy changes and no phase transition, takeHere $c_w$ is the water's specific heat capacity at constant pressure, $C_{\mathrm{cal}}$ is the participating vessel, heater and sensor heat capacity, $Q_{\mathrm{loss},\mathrm{out}}$ is signed outward heat leakage, and $W_{\mathrm{stir},\mathrm{on}}$ is stirring work entering the assembly. The storage term approximates its enthalpy change; for a rigid fixed-volume receiver use the corresponding heat capacity at constant volume. Calibrate $C_{\mathrm{eff}}$ independently if the purpose is to test electrical input against thermal storage. Using the same heating run to define $C_{\mathrm{eff}}=E_{\mathrm{el}}/\Delta T$ and then claiming it verifies the equality would be circular.

For the synthetic trace, stipulate $V=5\mathrm{V}$, $I=1\mathrm{A}$, $R=5\Omega$, $C_{\mathrm{eff}}=500J/K$, room temperature $20{}^{\circ }\mathrm{C}$, and loss coefficient $G=0.5W/K$, with negligible stirring input. The heater runs for $120\mathrm{s}$, beginning at room temperature. The calorimeter heat-loss correction model isIt predicts $E_{\mathrm{el}}=600\mathrm{J}$, $\Delta T\approx 1.13080\mathrm{K}$, storage $565.398\mathrm{J}$ and outward loss $34.602\mathrm{J}$. Neglecting leakage undercounts deposited energy by about $5.77\%$; assuming no loss predicts $1.20\mathrm{K}$. After switching off, the excess temperature undergoes exponential decay with time constant $C_{\mathrm{eff}}/G=1000\mathrm{s}$. A cooling slope can estimate $G$ only with known $C_{\mathrm{eff}}$ and a credible single-temperature model.

Controls separate causes of a temperature rise. Check the electrical meters' zeros and thermometer calibration; keep water mass, immersion, lid and stirring fixed between runs. Record a heater-off stirred baseline to estimate sensor drift and stirring input. Repeat cooling runs over the same temperature range to estimate environmental heat transfer; a baseline taken only at room temperature cannot measure warming-dependent leakage. Vary electric current while keeping $R$ and initial thermal conditions comparable: corrected deposited energy should scale as $I^2$ for equal durations. An independent calorimeter calibration or known reference heat capacity makes this a quantitative test rather than a qualitative observation.

Finite sensor response, incomplete thermal equilibrium, changing $R(T)$ or heat capacity, evaporation, and imperfect loss corrections limit the inference. A final temperature alone cannot distinguish smaller electrical input from larger heat leakage or a larger participating heat capacity. Agreement tests the stated assembly and its energy accounting; it does not show that all electrical work in arbitrary devices becomes heat.

Historically, the Royal Society's archive description of James Prescott Joule's manuscript dated 13 October 1840 records a coil of wire in a jar of water, a thermometer and a galvanometer, and summarizes his inference that warming depends on electrical resistance and the square of electric current intensity. This electrical experiment precedes Joule's paddle-wheel experiment; the input here is electrical work, whereas the paddle supplies mechanical work. The catalogue, rather than the inaccessible manuscript images, supports this account. ^[ref]

A cascade of nuclear interactions and secondary hadrons produced when an energetic hadron traverses matter. Neutral-pion decay supplies an electromagnetic component. Nuclear breakup, unobserved low-energy particles and leakage affect the visible response of a hadronic calorimeter.

Heat-engine calorimetry measures heat transfers so that observed work can be compared with the correct energy input. Measuring motion alone cannot supply the denominator of thermal efficiency.

The institutional Laboratory 16: Stirling Engine describes a modern teaching arrangement: an ethanol lamp drives a Stirling engine coupled to an electrical output device, which supplies a resistor immersed in water in a calorimeter. The prescribed readouts include fuel mass consumed, the pressure-volume loop, electrical readings, and water temperature. The sequence separates fuel input, gas-cycle work, electrical output, and Joule heating of the water. Warming of the water is the final useful output; it is not the engine's rejected $Q_c$. The handout distinguishes warm-up from steady operation and asks students to compare the final warming with direct heating by the lamp. It gives a procedure, not a table of measured efficiencies. Its pp. 1–3 therefore establish an apparatus and measurement design without supplying a new experimental result.

For a proposed calibration, use a known input of electrical work to determine the effective heat capacity $C_{\mathrm{eff}}$ of the water, vessel and immersed parts; stir to make the measured temperature representative. A matched run with no input estimates exchange with the room. For constant $C_{\mathrm{eff}}$ and no phase transition, the resulting model iswhere $Q_{\mathrm{leak},\mathrm{out}}$ is the signed net heat lost to the room during the measurement. Here $E_{\mathrm{in}}$ is supplied electrical work when the resistor lies inside the calibration boundary; for a separate external heat input it is the received heat. These are proposed controls, not controls reported in that handout. Slow cooling, changing material properties and imperfectly matched room exchange limit this inference; the approach to thermal equilibrium and heat capacity matter as much as the thermometer reading.

To measure the actual engine-input $Q_h$, one must instrument its hot-side transfer separately, or close a calibrated energy balance around a specified heater boundary. A heater's electrical work includes changes in its stored internal energy and heat transfer that bypasses the engine. At periodic operation a cold-side flowing-water measurement can instead estimate ${\dot{Q}}_c=\dot{m}{c}_p(T_{\mathrm{out}}-T_{\mathrm{in}})$, if mass flow rate $\dot{m}$ and specific heat capacity at constant pressure $c_p$ are known, with negligible storage and other heat exchange in that measurement boundary. It measures only the rejection captured by that stream. Include all remaining transfers before using the residual to infer ${\dot{Q}}_h=P_{\mathrm{out}}+{\dot{Q}}_c$. This measurement discipline makes a comparison with Carnot's theorem meaningful.

A calorimeter that infers transferred heat from ice melting at a specified equilibrium temperature. The latent heat sets the proportionality after correcting for other thermal changes and extraneous melting.

Joule heating is the dissipative conversion of electrical work into internal energy in an electrical conductor. In a metal, the electric field supplies energy to charge carriers, which transfer it to the material. This makes an immersed resistor a useful calibrated heater and makes electrical resistance a source of losses in wiring. Wind-driven electric currents are a proposed source of heating of an inflated giant planet.

For a stationary, purely resistive Ohmic conductor, use the passive sign convention: $V$ is the terminal electric potential difference from the entry to the exit of positive electric current $I$. The absorbed electrical power and Ohm's law giveThe last two expressions assume $V=IR$ at the relevant material temperature. For constant electrical resistance and electric current over a duration $\Delta t$, the dissipated energy is $E_{\mathrm{diss}}=I^{2}R\Delta t$; otherwise use $E_{\mathrm{diss}}=\int V(t)I(t)dt$. In SI, $\mathrm{V}\mathrm{A}=\mathrm{W}=J/s$.^[ref]

Absorbed electrical power $VI$ is more general than Joule heating: a capacitor can store energy, a battery can change its chemical energy, and an electrical circuit can deliver mechanical work or emit light. Even a heated resistor need not show an indefinitely rising temperature: in a thermal steady state, its dissipated power leaves by heat transfer. With the heater inside a calorimeter boundary, electrical input crosses as work; the subsequent heat transfer to the room is a different term in the first law of thermodynamics.

A second indirect route makes the experimental distinction especially clear: a calorimeter can measure dissolution of a weighed salt into a known amount of water, but its final ions are hydrated, not separated gaseous ions. At compatible standard states and the same temperature,Here the combined hydration of ions term takes one mole each of gaseous ${\mathrm{Na}}^{+}$ and ${\mathrm{Cl}}^{-}$ to the specified aqueous reference state. Its sum suffices: dissolution calorimetry alone cannot independently assign single-ion hydration of ions enthalpies, or determine $L_{\mathrm{diss}}$ without another input. A finite-concentration measurement also requires dilution corrections before it can supply $\Delta H_{\mathrm{sol}}^{\circ }$.

In a simple constant-pressure solution calorimeter, an insulated vessel, stirrer and thermometer follow the temperature change after mixing. A calibrated effective heat capacity $C_{\mathrm{eff}}$ converts the corrected temperature change to heat. With no applied heating, negligible external heat transfer, and only pressure-volume work, the approximate molar result isHere $n$ is the amount of substance dissolved and $C_{\mathrm{eff}}$ includes the chemical solution and the vessel. Cooling means positive dissolution enthalpy, because the dissolving sample takes heat from its surroundings.^[ref]

An actual reference measurement used potassium chloride, not sodium chloride: a platinum-lined calorimeter with an adiabatic shield and a quartz-oscillator thermometer. Because dissolution absorbed heat, measured electrical heating prevented loss of shield control; the readouts were electrical input and the vessel/shield temperature record. Stirring was checked visually in a dummy vessel. The thermometer was compared with a calibrated resistance thermometer, and electrical calibration bracketed the reaction. The reported result was $+(17.584\pm 0.017)\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ at $298.15\mathrm{K}$, for one mole of potassium chloride in $500$ moles of water.^[ref]

Checks on residual water, particle size and final concentration, and agreement with a second calorimeter, constrained systematic errors. Imperfect shield tracking required heat transfer corrections. This experiment measures an enthalpy of solution, not a direct lattice energy; the hydration and dilution bookkeeping still has to be supplied. The full article could not be opened here; the inspected indexed text supports the stated apparatus and checks, without reconstructing its complete uncertainty budget.^[ref]

A physicist.^[ref]

Infer an unknown heat capacity by thermal contact with a calibrated receiver and a measured common final temperature. The energy lost by hotter components equals the energy gained by colder components, with the calorimeter and surrounding heat transfer included. For appreciably varying heat capacities, use temperature integrals rather than constant $mc\Delta T$ terms.

A long-baseline neutrino oscillation experiment using a Fermilab beam, near and far detectors, and magnetized iron/scintillator sampling calorimeters. A charged current muon neutrino event gives a negative muon and a hadronic shower; their reconstructed energies estimate the incident neutrino energy.

Temperature describes thermal state and determines the direction of spontaneous heat transfer. A quantity of heat is energy transferred because of a temperature difference. It depends on the process and is measured in energy units; it is not another name for temperature. Equal heat inputs need not produce equal temperature rises in different masses or substances.

For a specified process, the specific heat capacity isOver a modest range with nearly constant $c$ and no phase change, $Q=mc\Delta T$. For a gas the constraint matters: heat capacity at constant pressure and heat capacity at constant volume generally differ because expansion can do work.

In method-of-mixtures calorimetry, a body of known mass $m_s$ at $T_s$ is transferred into water of mass $m_w$ at $T_w$ in a calibrated calorimeter initially also at $T_w$. If the final uniform temperature is $T_f$, the vessel and immersed fittings have heat capacity $C_{\mathrm{cal}}$, and losses are negligible,ThusStirring, quick transfer, a lid, a temperature-time record and allowance for surrounding heat exchange improve the inference. Ignoring the calorimeter's heat capacity systematically misattributes part of the energy uptake.

An alternative is electrical calorimetry: measure $E=\int VIdt$ supplied to a heater and compare the temperature rise with a blank or separately calibrated vessel. At nearly constant properties, $E=(mc+C_{\mathrm{cal}})\Delta T+Q_{\mathrm{loss}}$. An ice calorimeter instead measures the mass melted at a known phase-change temperature, using $Q=m_{\mathrm{melt}}{L}_f$ with appropriate corrections. Steam condensation similarly uses latent heat. These methods infer transferred energy from different measurable effects; a thermometer reading alone does not measure a quantity of heat.

For the historical laboratory context, intense heating could be obtained from an electric arc between carbon electrodes and from an oxyhydrogen flame. The first converts electrical work into a very hot emitting region; the second releases chemical energy rapidly in combustion. An intense light or an ability to melt a particular solid establishes a useful physical effect, but does not by itself provide a calibrated numerical temperature.

The cold end could be reached by rapidly evaporating a volatile material and reducing the surrounding pressure. In Faraday's carbon-dioxide and ether cooling experiment, a bath of solid carbon dioxide and diethyl ether was placed under an air-pump receiver. Removing vapour promoted further cooling. Faraday's 1845 report gives an alcohol-thermometer reading near $-106{}^{\circ }\mathrm{F}$ in the atmospheric-pressure bath and $-166{}^{\circ }\mathrm{F}$, approximately $-110{}^{\circ }\mathrm{C}$, under reduced pressure. He explicitly noted thermometer lag, thickening alcohol and possible errors of several Fahrenheit degrees. These are dated reported estimates, not a claim to a modern cold-temperature record.^[ref]

Measurement must be separated from temperature production. A liquid-in-glass thermometer works only while its liquid, glass and graduation remain usable. A gas thermometer infers temperature from reproducible gas pressure or volume changes, with calibration and departures from ideal-gas behaviour accounted for. At high temperatures, suitable expansion or calorimetric pyrometers can be compared against known points, but material changes and extrapolation introduce error. For example, transferring a hot body of known thermal properties into a calorimeter estimates its starting temperature only after allowing for its heat capacity and heat lost in transfer. The very hottest arc regions are not safely assigned the reading of an ordinary mercury thermometer.

Faraday's published lectures demonstrate the carbon-electrode electric light and electrical heating of platinum. They provide contemporary apparatus context; the later reprint's date is distinguished in its bibliography entry.^[ref]

For direct conversion, descending masses can drive paddles in an insulated vessel of water. Measure the lost gravitational potential energy and the temperature rise, including the vessel's heat capacity and corrections for unwanted heat transfer. Repeated reversal of the paddles and fixed baffles inhibit simple bulk co-rotation, directing more of the input towards dissipative fluid motion. Joule's paddle-wheel experiment tests a quantitative mechanical-to-thermal balance, rather than merely observing that stirring can warm water.^[ref]

For conversion through electricity, mechanical work drives a generator, whose electric current passes through a resistor in a calorimeter. The measured electrical input is $\int VIdt$, or $I^{2}Rt$ for constant electric current and electrical resistance. Joule heating increases the internal energy of the resistor and surrounding water. Compare the generator's mechanical input with the total thermal gains, including generator and circuit losses if the system boundary contains them. A battery alone demonstrates electrical-to-thermal conversion; without accounting for a mechanically driven source it does not establish the requested mechanical intermediate chain.

Using the given value relative to water, the copper requiresWith the modern thermochemical calorie, $1\mathrm{cal}=4.184\mathrm{J}$, so the ideal required work isEquivalently it is about $203\mathrm{kgf}\mathrm{m}$ using standard gravity. A nineteenth-century numerical equivalent of heat could give a slightly different rounded result. The stipulated constant specific heat capacity is used here; an unprovided vessel heat capacity and environmental losses would add to the required input.

The defensible core of William Odling's statement is a transfer of energy: sunlight helps build reduced food material, and its subsequent oxidation can supply animal heat and work. Here solar force means solar energy, and eremacausis means slow oxidation, rather than a special vital agent. The historical wording, from his 1866 lectures, must be qualified where it identifies the fuel with the substance of the working muscle or contrasts plant and animal metabolism too absolutely.^[ref]

In oxygenic photosynthesis, light absorbed by chlorophyll supports conversion of carbon dioxide and water into reduced organic material, with release of oxygen. An overall account, written for one glucose unit, isThis summarizes several stages, rather than describing a single elementary chemical reaction. The evolved oxygen comes from water. The product can furnish starch, other carbohydrates and precursors for fat and protein; the latter additionally requires a source of nitrogen. A human obtains these materials directly from plants or through animals that have eaten plants. Thus a grain meal and the flesh of a grain-fed animal can trace much of their chemical energy to the same original sunlight.^[ref]

The reverse overall balance for complete aerobic utilization of glucose isSimilarly a representative fatty acid, palmitic acid, has the complete oxidation balanceBoth equations conserve each chemical element. The relevant energy difference belongs to the complete set of reactants and products, including oxygen; merely breaking a bond does not supply net energy. In modern terms, controlled cellular respiration couples part of the available chemical free energy to ATP formation. ATP hydrolysis can then support muscle contraction, active transport and biosynthesis; some energy is transferred to heat. The process occurs through enzyme-mediated steps, without a flame, although complete combustion in a calorimeter can help measure the corresponding chemical energy change.

Glands and muscles continually replace constituents, and their proteins can be degraded. Amino acids can contribute carbon skeletons to energy metabolism, while their nitrogen is disposed of largely as urea. But ordinary muscle contraction does not require proportional destruction and rebuilding of the contractile protein at every stroke. Carbohydrates and fats are major fuels for muscle, and a gland can use those fuels to support secretion. The historical picture of burning the organ's own substance is therefore an inadequate general explanation.^[ref]

This correction was already an experimental issue before the exam. Edward Frankland's 1866 discussion compared the mechanical work of climbing with the energy that could be assigned to protein destruction estimated from urinary nitrogen. The latter was insufficient to account for all the work. His combustion measurements also distinguished the heat from burning a sample completely from the smaller amount available when nitrogenous material leaves the animal as still-combustible urea. Measuring external work, urinary products and calorific values gives the hypothesis a quantitative test; it is more informative than simply pointing out that an active animal gets warm.^[ref]

Finally, plants themselves carry out cellular respiration, and not every ecosystem obtains its input from sunlight: chemolithotrophy is an alternative. The plant-animal contrast is useful for following energy through an ordinary food chain, but it is not an exhaustive division of biochemical capabilities. Conservation of energy also requires the bookkeeping formfor an idealized interval in which $E_{\mathrm{food}}$ is the metabolizable food input, with energy lost in excreta already subtracted and other material-energy exchanges neglected. Here $W_{\mathrm{external}}$ is external work, $Q_{\mathrm{outward}}$ is net outward heat transfer, and $\Delta E_{\mathrm{body}}$ includes changes in stored chemical and other energy. The living body transforms energy; it does not create it.

supersymmetry pairs bosonic and fermionic degrees of freedom; broken supersymmetry permits partners heavier than known particles. At the Large Hadron Collider, strong production of squarks and gluinos is an important channel. If R-parity is conserved, partners are produced in pairs and their cascades end in a stable lightest supersymmetric particle. A neutral weakly interacting lightest supersymmetric particle escapes, giving missing transverse momentum together with jets and possibly leptons. Conservation is tested transversely because the initial parton longitudinal momenta are unknown.

The experimental observable isneutrinos produce genuine missing transverse momentum, and calorimeter mismeasurement or detector defects can imitate it. Searches use jet multiplicity, transverse momenta, missing momentum, angular separation cuts and a collider effective mass such as $m_{\mathrm{eff}}=p_T^{\mathrm{miss}}+\sum p_T^{\mathrm{jet}}$. Isolated-lepton selections explore complementary cascades. Trigger efficiency, jet energy scale, collision pileup and event cleaning affect acceptance.

A credible search estimates backgrounds from particle-physics control regions before inspecting the signal region. Invisible $Z\to \nu \bar{\nu }$ plus jets can be constrained using photon-plus-jets or visible $Z\to {\ell }^{+}{\ell }^{-}$ samples with control-region transfer factors. Lepton-enriched samples constrain $W$ and top backgrounds with lost leptons; multijet samples constrain fake missing momentum. A joint likelihood includes these observations and nuisance parameters for systematic uncertainties. Agreement with background gives upper bounds on signal rate and conditional mass exclusions. The pre-examination ATLAS 7 TeV analysis is a concrete example of this strategy and reported no significant excess.^[ref]

Electroweak production motivates multilepton searches; compressed spectra can give soft visible decay products and require special selections such as recoil against initial-state radiation. R-parity violation can instead remove the stable missing-particle signature, requiring other searches. An exclusion always depends on masses, branching fractions, production rates and decay assumptions; a null jets-plus-missing-momentum result does not exclude supersymmetry as a whole.

Strike a target with high-energy protons, collect negative pions and kaons with magnetic horns, and let them decay in a long evacuated decay tunnel: ${\pi }^{-}\to {\mu }^{-}{\bar{\nu }}_{\mu }$ and $K^{-}\to {\mu }^{-}{\bar{\nu }}_{\mu }$. Absorb residual hadrons and range out muons in shielding. Focusing selects the negative parent sign and therefore enriches muon antineutrinos; other parent decays leave a finite contamination. Monitor the parent flux and use a near detector to constrain the unoscillated spectrum.

A massive segmented calorimeter supplies interaction probability and measures the hadronic shower. Alternate dense absorber plates with scintillation counters or tracking planes, and magnetize the iron or add a downstream magnetic muon detector. The signal is a positive muon plus a hadronic shower; the sign rejects much of the muon neutrino contamination. For a contained muon, its range gives energy; for a penetrating track, curvature gives $p_{\perp }[\mathrm{GeV}]=0.2998B[\mathrm{T}]R[\mathrm{m}]$. Multiple measurements and a field map determine the fitted momentum and sign. Summed calibrated shower response estimates the struck quark's visible hadronization energy, not the energy of a free observable quark. Correct for invisible nuclear losses, noncompensating response, leakage and acceptance. Neglecting target recoil corrections, $E_{\nu }\simeq E_{\mu }+E_{\mathrm{had}}$ and $y\simeq E_{\mathrm{had}}/E_{\nu }$.

The Fermilab primary proton beam strikes a target, producing secondary pions and kaons. Horn focusing selects positive parents; their decays ${\pi }^{+}\to {\mu }^{+}{\nu }_{\mu }$ and $K^{+}\to {\mu }^{+}{\nu }_{\mu }$ produce a predominantly muon neutrino beam. A decay tunnel permits decay before the remaining charged secondaries encounter an absorber. A near detector constrains the flux and interaction spectrum, while the distant MINOS detector measures its change.

The alternating scintillation counters and magnetized iron give a sampling calorimeter and a magnetic tracking volume. A charged current ${\nu }_{\mu }N\to {\mu }^{-}X$ event has a long negative muon track and a localized hadronic shower. Estimate $E_{\mu }$ by range if contained or magnetic curvature if exiting, and $E_X$ by the calibrated scintillator pulse-height sum. Reconstruct $E_{\nu }\simeq E_{\mu }+E_X$, with corrections for nuclear binding, invisible energy and detector response. The muon charge separates most wrong-sign muon antineutrino events; short showers without the track are primarily neutral current candidates. The source's $730\mathrm{km}$ baseline is used in the calculations below.