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An echinoderm is a marine animal in the group containing sea stars, sea urchins and sand dollars. Adult organization commonly has five-part radial structure, whereas larvae can have bilateral symmetry. The skeleton consists of internal mineralized elements.^[ref]

The restriction of a component of the angular momentum operator to discrete eigenvalues; the Stern-Gerlach experiment tested its magnetic manifestation.

A distribution of quantum state populations over atomic magnetic sublevels with a nonzero expectation value of the angular momentum operator vector.

A rank-$k$ spherical tensor operator has components $T_q^{(k)}$, $-k\le q\le k$, that transform under rotation like an angular momentum $k$. Equivalently, their commutators with the angular momentum operator satisfyFor a vector operator, the rank is one and a conventional choice is $T_0=V_z$, $T_{\pm 1}=\mp (V_x\pm i{V}_y)/\sqrt{2}$. The Wigner-Eckart theorem separates the matrix elements of a spherical tensor operator into Clebsch-Gordan coefficients and a reduced matrix element.

Within the dipole and sudden approximation, angle-resolved photoemission spectroscopy measures the electron removal spectral function multiplied by a transition matrix element. Finite instrumental energy and angular resolution convolve this product, and inelastic scattering adds a background. Peak positions reveal occupied electronic band structure, while intrinsic line widths contain lifetime information. Matrix elements depend on photon energy, polarization, orbital character, and geometry, so weak intensity does not by itself establish the absence of an electronic state. Equilibrium photoemission does not directly measure unoccupied bands far above the Fermi level.

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.