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The functional arrangement below separates the polarization analysis from the resonance selection. The shield prevents the direct source beam from overwhelming the detector; the off-axis detector accepts gamma rays re-emitted by the samarium scatterer.

The europium source emits into magnetized iron, whose magnetization is alternated. A samarium scatterer selects nuclear nuclear resonance fluorescence, favouring photons emitted along the daughter recoil. A shielded NaI(Tl) scintillation counter records the scattered nuclear line. Compare the background-subtracted line rates for the two iron magnetizations, and use the calibrated sign of the analysing power to identify the photon helicity.

The measured circular polarization was negative, compatible after angular acceptance corrections with negative neutrino helicity. By the angular momentum conservation argument, the selected photon helicity and neutrino helicity have the same sign. Thus the experiment concludesfor the emitted relativistic electron neutrino. Finite acceptance dilutes the measured polarization; it does not justify equating the raw count asymmetry to $-1$.^[ref]

Neglect the participating quark and lepton masses, so that the weak charged current selects a left-helicity $u$ and a right-helicity $\bar{d}$. Choose $+z$ along the proton beam in the partonic centre-of-momentum frame. Then the up quark moves along $+z$ with spin projection $-1/2$, while the down antiquark moves along $-z$ with helicity $+1/2$ and also has spin projection $-1/2$ along $+z$. ThusThe collinear two-body initial state has no orbital projection about its collision axis. In this approximation the produced W boson therefore has the indicated transverse polarization; its longitudinal spin state is absent. The unpolarized hadron beams contain other constituent spin states, but these do not participate in this massless charged-current subprocess.

In the final state the Positron has helicity $+1/2$ and the electron neutrino has helicity $-1/2$. Along the outgoing charged-lepton direction their combined angular-momentum projection is $+1$, because the oppositely moving neutrino has spin projection $+1/2$ on that axis. Finite masses, noncollinear production and sea contributions modify the ideal configuration.

Starting with the electron neutrino state in a, the survival probability amplitude isTaking its squared absolute value givesThis derives the supplied survival probability from the evolution operator rather than from a Hamiltonian entry. It also shows why the two written forms agree. For the stipulated phase and $\theta =22.5^{\circ }$, ${\sin }^{2}2\theta =1/2$, so at $L=79\mathrm{km}$,The phase decreases monotonically from $7.9$ to $0.79$ over the open interval $1<E<10$. Differentiating gives $d{P}_{ee}/dE=[7.9/(2E^2)]\sin (15.8/E)$. The interior maxima occur at $7.9/E=n\pi$ and the interior minima at $7.9/E=(n+1/2)\pi$. Enumerating those phases givesThe minimum just above $1\mathrm{MeV}$ must not be lost by rounding the lower endpoint. Neither excluded endpoint is an additional extremum.

For solar neutrinos propagating in the same vacuum model, the vastly longer baseline makes the oscillations extremely fine in energy, with the same bounds $1/2$ and $1$. The figure uses a deliberately schematic enlargement of the oscillation spacing to show those bounds. A $1\mathrm{MeV}$ detector energy resolution averages very many periods, so $⟨{\cos }^2\phi ⟩=1/2$ andThe detector averages event rates weighted by source spectrum, interaction cross section and response; the displayed constant follows when these vary slowly across each tiny oscillation period. Reactor production physically gives electron antineutrinos. In this two-flavor vacuum approximation their survival probability is identical; the positive matter-potential discussion in d concerns solar neutrinos.

For a positive electron neutrino matter potential and the normal ordering implicit in positive $K$, choose $0\le {\theta }_m\le \pi /2$ by continuous diagonalization of the matter Hamiltonian. With $D=[(\cos 2\theta -K\rho {)}^2+{\sin }^{2}2\theta {]}^{1/2}$, the quadrant is fixed byAt high density $\cos 2{\theta }_m\to -1$, soTaking a principal-value arctangent alone and concluding ${\theta }_m\to 0$ would select the wrong branch. The production state is the second matter propagation eigenstate, rather than literally the vacuum mass eigenstate while still inside the dense medium.

Assume the solar density decreases slowly enough for adiabatic matter-enhanced neutrino conversion, and neglect subsequent Earth matter effects. The state follows ${\nu }_{2m}$ continuously into the vacuum mass eigenstate ${\nu }_2$. Since $⟨{\nu }_e|{\nu }_2⟩=\sin \theta$,This is substantially below the phase-averaged vacuum result $0.75$. There is no interference between two populated vacuum mass eigenstates in this ideal adiabatic limit. More generally, after phase averaging, $P_{ee}=\frac{1}{2}+\frac{1}{2}\cos 2\theta \cos 2{\theta }_m$ at production if the evolution is adiabatic. In the high-density limit, a nonadiabatic level-jump probability $P_c$ instead gives $P_{ee}=(1-P_c){\sin }^2\theta +P_c{\cos }^2\theta$. Thus the supplied core-density condition alone does not guarantee the numerical Earth result: the usual solar adiabatic assumption is also needed. These are the stipulated two-flavor parameters, not a precision prediction for the measured three-flavor solar spectrum.

The lepton mixing matrix is unitary, so each row has unit norm. Explicitly its electron row obeyswhere $c_{ij}=\cos {\theta }_{ij}$ and $s_{ij}=\sin {\theta }_{ij}$. This is also the normalization needed for unit electron neutrino survival at $L=0$.

Use $\mathrm{Re}(1-e^{ix})=1-\cos x=2{\sin }^2(x/2)$ and define ${\Delta }_{ij}=({\varphi }_i-{\varphi }_j)/2=\Delta m_{ij}^{2}L/(4E)$ in natural units. The exact vacuum neutrino oscillation result becomesThe given mass-squared differences satisfy $|\Delta m_{12}^2/\Delta m_{23}^2|\simeq 0.027\ll 1$, so the two fast oscillation frequencies are close: ${\Delta }_{13}={\Delta }_{23}+{\Delta }_{12}$. Approximating their phases as equal combines their coefficients intowhile $4c_{12}^2{s}_{12}^2{c}_{13}^4=c_{13}^4{\sin }^{2}2{\theta }_{12}$. HenceThe complex CP phase cancels from every $|U_{ei}{|}^2$. The phase-replacement approximation is uniformly reliable for $|{\Delta }_{12}|\ll 1$, especially at short reactor baselines; closeness of the mass-squared differences alone does not guarantee close sine values at arbitrarily large $L/E$. At medium baselines the distinct fast frequencies can beat, so the exact three-frequency expression is preferable if their detailed pattern matters.

A Positron signal is correct for the reactor's antineutrinos, through inverse beta decay,A true electron neutrino charged current interaction instead produces an electron, for example ${\nu }_e+n\to e^{-}+p$. This corrects the source's loose use of “neutrino” for the reactor beam.

After slowing, the Positron annihilates with an electron, generally producing two $511$ keV photons. The prompt visible energy includes both the Positron kinetic energy and annihilation energy; neglecting the small neutron recoil,A delayed neutron capture supplies a powerful coincidence tag. Detecting only two annihilation photons discards useful prompt energy and does not uniquely identify the charged current event.

An experimental contributor to the Sudbury Neutrino Observatory’s study of solar neutrinos. A neutral current reaction measured the total active-neutrino flux, while a separate reaction preferentially measured electron neutrinos. Comparing the two showed that some neutrinos reached Earth in a different flavor, supplying direct evidence for solar-neutrino flavor transformation.^[ref]

Solar neutrino detection asks how an invisible electron neutrino from the Sun becomes a measurable laboratory record. The Homestake experiment stores a reaction product for later counting; Kamiokande records light promptly. The former measures an exposure-integrated rate, whereas the latter can retain event times and statistical source-direction information. These are complementary ways to connect stellar nuclear fusion with neutrino astronomy.

A detector samples a weighted part of a neutrino flux, rather than counting every passing neutrino. For a target population $N_T$, an ideal reaction rate iswhere ${\varphi }_e(E)$ is the differential electron neutrino neutrino flux and ${\sigma }_e(E)$ is the relevant cross section. A detection efficiency belongs inside the integral if it depends on energy. Two detectors can therefore disagree in raw count rate while observing the same source. Interpreting a deficit requires both a source prediction and a detector response.

The SNO Collaboration used $1000$ tonnes of heavy water in a $12\mathrm{m}$ acrylic sphere, surrounded by ordinary water shielding and $9456$ photomultipliers. Weak charged current (CC) breakup of a deuteron selects electron neutrinos; weak neutral current (NC) breakup measures all active neutrino flavors; electron elastic scattering (ES) has reduced non-electron sensitivity.^[ref]