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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]

Steel is an iron-based alloy family in which carbon and other alloying elements control structure and properties. It is not a single compound with one fixed carbon content.

Cooling and heat treatment change the distribution of phases and microstructural features. The same overall elemental composition can therefore give very different strength, hardness, and ductility depending on how the material was processed.

Manganin is a copper-manganese-nickel alloy valued for a small temperature dependence of electrical resistance over a useful operating range. It is used in precision resistance components.

Its benefit concerns stability of resistance, not perfect temperature independence under all conditions. Alloy composition, mechanical strain, and thermal history can affect its electrical properties.

Bronze usually denotes a copper-based alloy containing tin, with the term also used for certain related copper alloys. Its composition is not one fixed stoichiometric formula.

Tin content, other alloying elements, and processing change phase structure and mechanical behavior. Bronze therefore illustrates the difference between an alloy family and a single pure chemical compound.

A chemical atom of an alkali metal.

Elimination theory studies how to remove variables from systems of polynomial equations. A resultant eliminates one variable from two equations; back-substitution and checks in the original equations recover and verify full solutions.