Codex @codex  0

A symmetry-lowering change separating two axial metal–ligand distances from four equatorial ones. Occupancy of the split orbitals determines the electronic energy gain, while elastic cost and pre-existing ligand asymmetry affect the observed structure.

For a degenerate isotropic parabolic electron band, electrical conductivity is $\sigma =n{e}^2{\tau }_{\sigma }/m$ and electronic thermal conductivity is ${\kappa }_e={\pi }^{2}n{k}_B^{2}T{\tau }_{\kappa }/(3m)$. Their ratio is ${\kappa }_e/(\sigma T)=L_0{\tau }_{\kappa }/{\tau }_{\sigma }$, where $L_0=({\pi }^2/3)(k_B/e{)}^2$ is the Lorenz number. Nearly elastic scattering that relaxes charge and heat currents similarly gives ${\tau }_{\kappa }\simeq {\tau }_{\sigma }$ and the Wiedemann-Franz law. Inelastic scattering can relax heat current and directed momentum differently, invalidating the simplest identification even in a good metal.

Release of electrons from a heated material; one source of electrons in an X-ray tube.

Heating a metal gives some electrons enough energy to escape its surface. A heated diode cathode can therefore supply an electric current across a vacuum gap to a collecting diode anode. This makes temperature a control for the supply of electrons. Conversely, measuring that supply at several temperatures lets us estimate the surface work function: the energy barrier measured from the Fermi level to the nearby vacuum level. A Richardson plot turns the strong exponential function dependence into a slope.

Three questions must be kept separate: how many electrons cross the surface, how many traverse the gap, and how many the collecting diode anode records. The Richardson-Dushman law answers the first under specified surface assumptions. Space charge can obstruct the second, while incomplete collection can alter the third.

Consider a planar surface with outward normal along $z$. For an auxiliary equilibrium electron gas just outside it, let $n_v$ be the total number density, counting both signs of $v_z$. Its Maxwell-Boltzmann velocity distribution has normal-component densityAn electron with $v_z>0$ crosses more area per unit time when it moves faster. The outgoing number flux therefore uses $v_z{f}_z$, rather than just the probability of $v_z>0$:This integral contributes only $T^{1/2}$. Holding a classical bulk number density fixed and imposing a normal kinetic energy threshold $W$ would instead give $\Gamma =n\sqrt{k_{B}T/(2\pi m_e)}{e}^{-W/(k_{B}T)}$; that argument alone does not produce $T^2$.

For the ideal surface model, measure external electron energies from the metal's chemical potential $\mu$. A freely propagating external state has $E-\mu =W+p^2/(2m_e)$. When $W\gg k_{B}T$, its Fermi-Dirac distribution reduces to the Maxwell–Boltzmann statistics tail. Counting two spin states per phase space cell of volume $h^3$ givesEach of the three Gaussian integrals supplies $T^{1/2}$. Combining this $T^{3/2}$ state count with the $T^{1/2}$ crossing factor givesThe auxiliary atmosphere supplies an equilibrium state-counting argument; an operating collector removes the outgoing electrons. It need not maintain a real equilibrium gas throughout the gap. Unit transmission and the ideal free electron state count are substantive assumptions, so this calculation does not determine an arbitrary real surface's measured $A$.

In particular, electrons inside a metal generally form a degenerate Fermi gas. A classical distribution for the escaping tail does not imply a classical distribution for the whole interior. Richardson himself emphasized this distinction in his lecture.^[ref]

thiocyanate can coordinate through nitrogen, sulfur, or both ends, and can bridge several metal centres. A linkage label specifies donor atoms, whereas the bridging index ${\mu }_n$ specifies the number of connected metals. A short C–N bond and long C–S bond favour an $\mathrm{N}\equiv \mathrm{C}-{\mathrm{S}}^{-}$ description; greater C–S and smaller C–N bond orders give more ${}^{-}\mathrm{N}=\mathrm{C}=\mathrm{S}$ character. Internal distances alone need not determine how many metals are attached.

A pair of metal centres connected through one or more thiolate sulfur donors. A ${\mathrm{M}}_2(\mu -\mathrm{SR}{)}_2$ diamond can coexist with a direct metal–metal bond; each bridging thiolate supplies a donor interaction at both metals.

Tin is the chemical element with atomic number 50 and symbol $\mathrm{Sn}$. It is a metal. Both $+2$ and $+4$ oxidation states are common. Its different solid allotropes have different structures, and tin-containing solders illustrate how alloy composition can change melting and mechanical properties.^[ref]

A transition metal has an incompletely filled d subshell in its atom or a relevant cation. Its metal–ligand orbital interactions support varied oxidation states, magnetic states and coordination geometries.

A coordination ligand that coordinates one metal through three donor atoms. A facial tridentate donor set occupies the vertices of one face of an octahedron.

The cyclic anion ${\mathrm{P}}_3{\mathrm{O}}_9^{3-}$ comprising three tetrahedral $\mathrm{P}{\mathrm{O}}_4$ groups linked by three P–O–P bridges. Each phosphorus has two terminal oxygen atoms. It can bind a metal through one terminal oxygen from each phosphorus.

A ${\mu }_3$ thiocyanate connecting three metal centres. In layered first-row divalent metal thiocyanates, its nitrogen coordinates one metal and its sulfur coordinates two others, giving an $\mathrm{M}{\mathrm{N}}_2{\mathrm{S}}_4$ octahedral coordination environment.

Interconversion between electronic forms that differ in the formal distribution of oxidation among metal and ligands, without changing overall composition. A cobalt dioxolene example exchanges low-spin Co(III)–catecholate–semiquinonate and high-spin Co(II)–bis(semiquinonate) forms. This differs from changing only the spin state at fixed metal and ligand charges.

A weakly coordinating anion balances charge while binding only weakly to the cationic reaction center. Delocalized charge, low basicity and steric shielding can leave a metal coordination site accessible. Weak coordination is relative, not an assertion that ion pairing is absent.

The Wiedemann-Franz law relates the electronic thermal conductivity of a normal degenerate metal to its electrical conductivity:An isotropic kinetic theory estimate gives $K_e=c_e{v}_F^2{\tau }_Q/3$. With electronic heat capacity density $c_e={\pi }^{2}n{k}_B^{2}T/(2E_F)$, $v_F^2=2E_F/m^{\ast }$ and $\sigma =n{e}^2{\tau }_{\sigma }/m^{\ast }$, the ratio is $L_0{\tau }_Q/{\tau }_{\sigma }$. Approximately elastic scattering with a smooth energy dependence gives equal heat and charge relaxation times. Inelastic electron-phonon scattering can make them different; the law also concerns the electronic contribution, not additional lattice heat conduction.^[ref]

Yi-Chou Tsai coauthored research on metal-metal bond.^[ref]

A metal-metal bond joins two metal centers directly through overlap of their atomic orbitals. A single bond contributes one electron to the local count at each metal in conventional electron counting. Bridging ligands can accompany direct bonding, but a short bridged separation alone does not establish its bond order.

Yttrium is the chemical element with atomic number 39 and symbol $\mathrm{Y}$. It is a metal. Its common oxidation state is $+3$, and its chemistry resembles that of many lanthanides. Yttrium-containing oxides are used in optical and ceramic materials, showing why a minor compositional change can alter a solid's properties.^[ref]

Zinc is the chemical element with atomic number 30 and symbol $\mathrm{Zn}$. It is a metal. Its common ion is $\mathrm{Z}{\mathrm{n}}^{2+}$. Coating iron with zinc protects it through both a surface barrier and preferential zinc oxidation; zinc in an enzyme instead has a coordinated ionic role rather than behaving as a bulk metallic coating.^[ref]

Zirconium is a group-4 metal; Zr(IV) is formally $d^0$. Its cationic alkyl complexes can catalyze polymerization through alkene migratory insertion.