For emission angle $\theta$ measured from the surface normal, a free outgoing photoelectron has parallel wavevector $k_{\parallel }=\sqrt{2m_e{E}_{\mathrm{kin}}}\sin \theta /\hslash$. Translation symmetry parallel to a clean periodic surface conserves this component modulo a surface reciprocal lattice vector, after accounting for photon momentum when needed. With initial binding energy $E_B$ below the Fermi level, its kinetic energy immediately outside the sample is $E_{\mathrm{kin}}=h\nu -{\varphi }_s-E_B$, where ${\varphi }_s$ is the sample work function. This sample-vacuum energy is distinct from the analyzer-referenced energy when contact potentials are present. Perpendicular crystal momentum generally needs a final-state model, whereas a strictly two-dimensional band has no perpendicular dispersion to reconstruct.

The angle subtended by an object at an observer. For a circular object of diameter $D$ viewed face-on at a distance $d$ much larger than $D$, its angular diameter is approximately $D/d$ in radians.

A direction described by one or more angles relative to a specified reference.

The Geiger-Nuttall law expresses the extreme sensitivity of an alpha decay half-life to its Q value. In a restricted range of heavy nuclei, a useful local form is ${\log }_{10}[T_{1/2}/T_{\ast }]\simeq a{Z}_d/\sqrt{Q_{\alpha }}+b$, where $Z_d$ is the daughter atomic number, and the constants depend on the chosen units and nuclear family. It is named for Hans Geiger and John Mitchell Nuttall.

The Coulomb barrier outside the daughter has $V(r)=2Z_d\alpha /r$ in natural units. Its outer turning point is $r_t=2Z_d\alpha /Q_{\alpha }$. The WKB approximation gives a penetration probabilitywhere $\mu$ is the daughter-alpha reduced mass. The leading large-barrier exponent is $2\pi (2Z_d\alpha )\sqrt{\mu /(2Q_{\alpha })}$, producing the approximate $Z_d/\sqrt{Q_{\alpha }}$ dependence. Finite nuclear size modifies the exponent and the slowly varying intercept. The nuclear decay constant also contains an assault frequency and an alpha-cluster formation probability. Extra orbital angular momentum raises the barrier further, and odd-nucleon structure can introduce substantial hindrance. Thus extrapolation from even-even decays gives an order-of-magnitude reference rather than a precision prediction for an odd-odd nucleus.

The ground state of an even-even nucleus has $J^P=0^{+}$, and the emitted alpha particle also has $0^{+}$. In an alpha decay to daughter spin $J_Y$, addition of angular momentum permits total zero only when the relative orbital angular momentum is $L=J_Y$. Parity conservation gives $+1={\pi }_Y(-1{)}^L$, hence the daughter has natural parity, $0^{+},1^{-},2^{+},3^{-},\dots$. This rule does not itself restrict $J_Y$ to even integers. An even-$J$ sequence such as $0^{+},2^{+},4^{+}$ requires additional nuclear structure, for example a positive-parity ground-state nuclear rotational band.

In two-body alpha decay of a parent at rest, momentum conservation makes daughter and alpha momenta equal and opposite. The nonrelativistic kinetic energies therefore satisfy $T_d/T_{\alpha }=m_{\alpha }/M_d$, givingAn excited daughter reduces the available Q value. To leading recoil order, an alpha line separation $\Delta T_{\alpha }$ corresponds to excitation $E_x=(1+m_{\alpha }/M_d)\Delta T_{\alpha }$. Thus ratios of excitation energies inferred from line separations do not need the common recoil factor. These formulas assume kinetic energies small compared with rest masses; an exact relativistic two-body decay calculation can replace them when necessary.

A nickel–iron alloy with unusually small thermal expansion over a limited temperature range. Composition and treatment matter; its name does not mean every specimen has zero expansion. The Guillaume thermal expansion experiment explains how the effect was measured.^[ref]