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A binary crystal structure with one species at the corners of a primitive cubic unit cell and the other at its body centre. Each species has coordination number eight. Distinct chemical labels mean that the body-centre translation is not a translation symmetry; its Bravais lattice is primitive cubic, not body-centred cubic.

How can a clear chemical solution reveal otherwise invisible mobile ions? The documented Conductivity of Solutions by Harvard Natural Sciences Lecture Demonstrations demonstration uses metal electrodes enclosed in tubing, an electrical supply, and a lamp in the circuit. The protocol specifies about $150\mathrm{mL}$ samples in labeled $250\mathrm{mL}$ beakers and a separate deionized-water rinse station. The recorded readout is lamp brightness after the electrodes enter each sample; the simplified variant tests water before and after adding sodium chloride. These are apparatus and procedure specifications, not new measurements reported here.^[ref]

The useful comparison is water, a sucrose chemical solution, and a sodium chloride chemical solution with the same probe geometry. A documented university account reports no visible light with water, a glowing lamp with dissolved sodium chloride, and a weak filament glow with its sucrose sample. The diagram summarizes the latter as little or no visible light; the precise detection threshold depends on the tester. Sucrose is a control for the idea that any dissolved material must conduct: its dispersed neutral molecules do not provide the abundant mobile ions supplied by the salt. This contrast is a documented teaching observation, not a numerical calibration of lamp brightness.^[ref][ref]

Rinsing the electrodes between samples controls carryover of dissolved ions, and the water blank checks the solvent background. Repeating the blank after the salt sample would test whether that rinse was effective. Additional methodological controls are fixed immersed area, electrode separation, supply setting, temperature, and comparison at stated concentrations. These additional controls are proposed checks, not undocumented claims about the source demonstration.

The inference is that mobile electric charge carriers complete the circuit. In the chemical solution, an electric field drives positive ions along the field and negative ions against it; multiplying each drift velocity by its electric charge makes both contributions to conventional current point the same way. With an alternating supply the direction reverses each half-cycle. The external wires instead carry electric current through electrons. The lamp alone identifies neither the species nor an exact electrical conductivity.

Important limits remain. Water has a small intrinsic population of hydronium and hydroxide, so a dark lamp does not prove absence of ions. Changes in probe geometry affect measured electrical conductance; electrolysis, electrode interfaces, contamination, and heating can also affect a real circuit. The comparison supports a transport model but does not independently measure a coordination number, a partitioned partial atomic charge, or a lattice energy.

Nor does poor conduction by ordinary dry sodium chloride establish that all ionic crystals are electrical insulators. Crystallographic defects can allow transport, and solid electrolytes can have substantial ionic electrical conductivity without melting. The authored abstract of A lithium superionic conductor by Noriaki Kamaya and others reports room-temperature ionic conduction in a solid electrolyte; the full experimental article was not accessible here.^[ref]

A discrete molecular assembly in which metal–ligand connections surround a cavity. coordination number, preferred binding angles and ligand connectivity restrict possible cage stoichiometries.

The spatial arrangement of donor atoms around a central atom. coordination number counts neighbours but does not by itself distinguish geometries with the same number of donors.

In this structural use, the coordination number counts nearest neighbouring constituents. State which neighbours are counted and which ionic crystal structure is being described; it is not an ion's electric charge number.

For two point electric charges in vacuum, Coulomb's law gives the interaction potential energywith zero at infinite separation. Here $r_{ij}$ is the distance between them and ${ϵ}_0$ is the vacuum permittivity. Opposite signs give negative potential energy and attraction; equal signs give positive potential energy and repulsion. For a finite collection of distinct point electric charges, sum over $i<j$ to count each pair once. A bulk ionic crystal requires the extended lattice sum, including both signs, rather than multiplying one attractive pair by the coordination number.^[ref]

At short distances, overlapping electron density makes the simple point-charge picture inadequate. Quantum mechanics, including the Pauli exclusion principle, matters to the steep repulsive contribution. Pure attraction proportional to $-1/r$ would collapse as $r\to 0$; it cannot set an equilibrium spacing on its own. A useful illustrative model isHere $r$ scales an assumed fixed arrangement, $A$ packages its attractive contribution, and $B/r^n$ represents short-range repulsion. The constants have units energy times length and energy times length to the power $n$, respectively. This is a model, not a fitted description of a named salt.^[ref]

Differentiating gives $r_0=(nB/A{)}^{1/(n-1)}$ and $U^{\prime \prime }(r_0)=(n-1)A/r_0^3>0$, so the stationary spacing is a stable minimum. For the diagram's stipulated $n=9$, put $x=r/r_0$ and $E_0=A(1-1/n)/r_0>0$. ThenThe dimensionless curves show how competing interactions select a spacing. Their energy scale and exponent are synthetic; neither is a measured lattice energy or a melting prediction.

An $A{B}_2$ crystal structure with A on a face-centered cubic lattice and B occupying all eight tetrahedral sites of its conventional cubic unit cell. The coordination numbers are eight for A and four for B. The cubic cell contains four $A{B}_2$ formula units.

A geometric heuristic favouring the largest coordination number compatible with neighbouring opposite-charge ions touching the central ion without overlap. A limiting radius ratio rule comes from additionally making the surrounding ions just touch. It is a hard-sphere model, not an exact stability criterion.

A hydration shell is a solvation shell in water. The existing hydration of ions discussion describes preferential orientations around cations and anions.

Ordinary solid sodium chloride has the rock salt structure. Choose a conventional cube of side $a$ with chlorine anions at the corners and face centers, and sodium cations at the edge centers and cube center. Corner sites are shared by eight cubes, face sites by two and edge sites by four. The unit cell counts are thereforeThere are four formula units per conventional unit cell, rather than four separate NaCl molecules.^[ref]

The Bravais lattice is a face-centered cubic lattice. One choice of primitive lattice translation vectors and atomic basis isThe determinant gives a primitive unit cell volume of $a^3/4$; that cell represents one cation and one anion. Four such volumes make the conventional cube. The union of all ion positions resembles a primitive cubic lattice of step $a/2$ only if the chemical labels are discarded: translation by $a(1,0,0)/2$ exchanges unlike ions and fails to preserve the chemical structure.

Each ion has six nearest unlike neighbors at distance $a/2$, along the positive and negative directions of the three cube axes. Thus the coordination number is six. Within either same-species face-centered cubic lattice, the nearest same-ion neighbors instead number twelve: their displacements are permutations of $(\pm a/2,\pm a/2,0)$, all of length $a/\sqrt{2}$. Since $a/\sqrt{2}>a/2$, those twelve are a more distant shell in the complete sodium chloride structure. The number twelve describes the same-ion FCC lattice neighborhood; six describes the nearest unlike-ion neighborhood.

For the usual octahedral Ni(II) hydration-isomer series, distinguish coordinated water from water of crystallization:The first two coordination spheres have no cis/trans choice because they contain zero or one chloride. The last hydration class has two geometrical isomers: the chlorides occupy adjacent or opposite octahedral positions. Thus there are three hydration classes and four structures when cis/trans isomerism is included. Every formula retains two chlorides and six waters overall; the coordination number remains six. The schematic in B6(c) displays the distinct octahedral connectivities. This enumeration states the usual six-coordinate interpretation, rather than treating unspecified changes in coordination number as further members of the same elementary series.

CO and CO${}_2$ are discrete molecules, whereas the other oxides gain stability from extended bonding or ionic lattices. carbon's compact 2p orbitals overlap effectively with oxygen 2p orbitals, allowing strong multiple bonds that satisfy the valence requirements within small molecules. Intermolecular forces between CO or CO${}_2$ molecules are much weaker than those internal bonds, so low condensation temperatures and gaseous states at room conditions are unsurprising. CO${}_2$ is nonpolar overall but still has London dispersion forces and interactions between electric quadrupoles; “nonpolar” does not mean no attraction.

For larger Group 14 atoms, overlap with oxygen 2p orbitals is less effective for strong localized multiple bonding. SiO${}_2$ instead forms an extended network of SiO${}_4$ tetrahedra with each oxygen bridging two silicon atoms. Breaking or melting the network involves much more than separating intact SiO${}_2$ molecules. The Si–O electronegativity difference is $3.61-1.92=1.69$, so these are strongly polar covalent bonds rather than evidence for a molecular solid.

For Pb–O the difference is $3.61-1.85=1.76$, and ionic Pb${}^{2+}$/O${}^{2-}$ or Pb${}^{4+}$/O${}^{2-}$ descriptions provide useful starting points, with appreciable covalent contributions. PbO has layered structures with asymmetric oxygen coordination consistent with a stereochemically active 6s${}^2$ pair. A common PbO${}_2$ structure is rutile-like, with PbO${}_6$ octahedra linked through oxygen. Extended lattice interactions account for their solid states.^[ref] The similar Si–O and Pb–O electronegativity differences show why a single difference threshold cannot determine the entire bonding type, coordination number or structure.

Two bidentate bipyridine ligands provide four N donors and three terminal NCS ligands provide three more. Adding one water gives $n=1$ and coordination number eight:Eight-coordinate arrangements, distorted between familiar ideal polyhedra, are plausible for a sterically crowded Eu(III) centre. A second water would give coordination number nine, which is not forbidden for a lanthanide; the requested value is a chemically motivated prediction, not something fixed by charge balance. Diffraction or a direct hydration-number measurement is needed to establish it uniquely.

The bipyrimidine ligand has two distinct N,N-chelating pockets. It can use one pocket as a terminal bidentate ligand, or use both to bridge two Eu centres. Take all three nitrates per Eu to be bidentate O,O donors.

A consistent structure for B is $\mathrm{Eu}(\mathrm{N}{\mathrm{O}}_3{)}_3(\mathrm{bpm}{)}_2$, with two terminal, singly chelating bpm ligands. Six nitrate oxygens plus four bpm nitrogens give coordination number ten. The unused pocket of each terminal bpm points away from that Eu centre.

For C, use a dimer $\mathrm{E}{\mathrm{u}}_2(\mathrm{N}{\mathrm{O}}_3{)}_6(\mathrm{bpm}{)}_3$. Place one bpm between the metals, chelating each Eu through a different N,N pocket. Add one terminal bidentate bpm to each Eu and three bidentate nitrates to each Eu. Each metal again has $2+2+6=10$ donor atoms, while the overall Eu:bpm ratio is $2:3$. In this model all four nitrogens of the bridging bpm are used, but only two of each terminal bpm are used.

Both structures have their accessible coordination sites occupied by multidentate ligands, and the chelate effect favours retention of those donors over adding or exchanging a solvent molecule. This accounts for the absence of coordinated solvent while allowing a higher coordination number than the thiocyanate example. Stoichiometry alone does not exclude polymeric alternatives or different nitrate binding modes; the monomer/dimer pair is an explicit internally consistent prediction.

$M^1=\mathrm{Li}$ and $M^2=\mathrm{K}$. Both metals are bridged by the two diisopropylamide nitrogen atoms in a four-membered Li–N–K–N ring. The smaller lithium ion accommodates the two amides plus one THF oxygen, while the larger potassium ion has space for two THF donors and more readily adopts the higher coordination number. The principal Li–N and Li–O contacts should be shorter than their potassium counterparts because of the much smaller lithium ionic radius. A larger number of donors does not imply shorter individual K–donor bonds.

The twelve face-centred cubic neighbours are $(a/\sqrt{2})(\pm 1,\pm 1,0)$ and permutations of the zero coordinate. Eight vectors have a nonzero given component, each contributing $a^2/2$, while off-diagonal terms cancel between signs. ThusFor a cubic neighbour shell this can also be checked from ${\sum }_j{r}_{j,\alpha }{r}_{j,\beta }=(z{a}^2/3){\delta }_{\alpha \beta }$ with coordination number $z=12$.

The large rises mark the opening of Ga and As K-shell photoionization channels at their respective thresholds. Above each edge, the outgoing photoelectron interferes with waves backscattered by neighboring atoms; that interference modulates the absorption transition probability. Near-edge resonances and multiple scattering require a different level of treatment from the simple high-energy single-scattering formula. Ga has four As nearest neighbors, while As has four Ga nearest neighbors, so the two edges sample different central atoms and scattering factors even though they probe the same nearest-neighbor bond lengths.

Write $k=\sqrt{2m_e(E-E_0)}/\hslash$ and $\chi =(\mu -{\mu }_0)/{\mu }_0$ relative to a smooth atomic background. The phase $2k{R}_j+{\varphi }_j(k)$ includes the outward-and-return propagation distance and atomic phase shifts. The factor $R_j^{-2}$ describes geometrical spreading; the backscattering amplitude depends on the neighbor species. The exponential $e^{-2R_j/\lambda (k)}$ accounts for loss of coherent photoelectron propagation. Averaging over Gaussian bond-length fluctuations gives $⟨e^{2ik\delta R}⟩=e^{-2k^2{\sigma }_j^2}$.

The printed polarization factor needs a notation correction: for K-shell dipole absorption it is proportional to $({\hat{\mathbf{R}}}_j\cdot \mathbf{ϵ}{)}^2={\cos }^2{\theta }_j$, not literally the cosine of a dimensional vector dot product. Four equivalent tetrahedral bonds have an isotropic sum of these projection weights. If the sum is over shells rather than individual neighbors, it also needs the coordination number. A common conventional amplitude contains $S_0^2{f}_j(k)/(k{R}_j^2)$; omitted normalization and $1/k$ factors must be absorbed in the printed $f_B$ convention rather than silently changing its dimensions.

These are the ingredients of extended X-ray absorption fine structure. A shell fit uses calculated or reference scattering amplitudes and phases, and refines bond distances, coordination and mean-square relative disorder together with an edge-energy correction. Fourier transformation separates approximate distance contributions, but the uncorrected peak position is shifted by the scattering phase and is not an exact bond length.^[ref]

Sample A has the larger Ga–As bond length. Its oscillations advance slightly faster with $k$: at higher $k$ its corresponding maxima lie to the left of B's. Reading several full cycles gives approximate periods $\Delta k_A\simeq 1.47{\mathring{\mathrm{A}}}^{-1}$ and $\Delta k_B\simeq 1.50{\mathring{\mathrm{A}}}^{-1}$. In the constant/common-phase-slope estimate, $R\propto 1/\Delta k$, soThis is a rough percent-level difference, not a precise fitted strain. Nonconstant atomic phase shifts affect the absolute distance inferred from a period and also limit this fractional estimate. The graph does not establish which alloy has which indium concentration or how much its bulk lattice parameter changed. Local Ga–As relaxation need not follow a uniform bulk strain exactly.

Low temperature reduces thermal relative motion, preserving high-$k$ oscillations and separating small phase changes more clearly. It does not remove zero-point motion or static alloy disorder. To obtain an illustrative disorder estimate, take the high-$k$ envelope of the plotted $k\chi (k)$ to fall from about $0.19$ at $k=7.5{\mathring{\mathrm{A}}}^{-1}$ to $0.12$ at $k=10.4{\mathring{\mathrm{A}}}^{-1}$. If other envelope factors vary slowly over this interval,The value is only an order-of-magnitude, model-dependent reading: the backscattering amplitude, mean free path and normalization have not been independently removed. In particular, ${\sigma }^2$ is the bond-projected mean-square relative displacement,including correlations and static disorder. It is not uniquely either atom's mean-square displacement. Equal uncorrelated projected displacements would each contribute ${\sigma }^2/2$, but that extra assumption is not supplied by the figure.

Taking Cu K$\alpha$ wavelength $\lambda =0.15418\mathrm{nm}$, Bragg's law gives $d=\lambda /(2\sin \theta )$. For cubic $111$, $a=\sqrt{3}{d}_{111}$ and the nearest Zn–S separation is $\sqrt{3}a/4=3d_{111}/4$. For ideal wurtzite structure, $c=2d_{002}$ and the nearest separation is $3c/8=3d_{002}/4$. ThereforeBoth have tetrahedral nearest-neighbour coordination number four, but different further-neighbour stacking and lattice relaxation can shift their equilibrium bond lengths. The hexagonal value assumes ideal $c/a=\sqrt{8/3}$ and $u=3/8$; a real distorted wurtzite structure can have unequal axial and inclined bonds. One measured $002$ spacing alone does not determine every bond length without that structural assumption.

Each Ti has coordination number six, with oxygen neighbours forming a distorted octahedron. Each oxygen has coordination number three, with titanium neighbours forming a planar triangle. For oxygen at $(u,u,0)$, the nearest Ti positions can be taken as $(0,0,0)$ and $(1/2,1/2,\pm 1/2)$; all three lie in the plane $x=y$. The triangle is not generally equilateral. Equal Ti–O bond lengths in the following calculation do not make every angle ideal.

The coordination number rises from six in the rock salt structure to eight in the CsCl structure. In the elementary hard-sphere ionic bonding model, accommodating eight opposite neighbours favours a larger cation/anion radius ratio: the geometrical limiting ratios are $\sqrt{2}-1$ for sixfold and $\sqrt{3}-1$ for eightfold coordination. The high-pressure change is therefore consistent with $r_K/r_{Cl}$ increasing under compression. Sincethe chloride ion is inferred to be more compressible than the potassium ion in this radius-ratio interpretation. This is a qualitative structural argument; real ionic radii depend on coordination and electronic structure, and the transition does not uniquely determine separate microscopic compressibilities.