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A four-membered titanium–oxygen–carbon–carbon intermediate in carbonyl olefination. Fragmentation yields a carbon–carbon double bond and a titanium–oxygen product.

A suitable structure is a triangular Ti${}_3$ metal cluster with one 1,2-Bis(dimethylphosphino)ethane chelate and one terminal chloride at each titanium, and one bridging ligand $\mu$-Cl spanning each edge. Include a direct Ti–Ti single metal-metal bond along each edge. The last panel distinguishes these direct bonds from the chloride bridges.

The six chlorides have total formal charge $-6$, so equivalent centers are Ti(II), $d^2$. At each titanium the two phosphorus donors supply four electrons, the terminal chloride supplies two, and the two incident chloride bridges each donate a pair in ionic electron counting. Two incident Ti–Ti single bonds supply another two electrons:Each $d^2$ center can contribute one electron to each of its two edge bonds, making three bonds from six metal electrons overall. This is a 14-electron early-metal cluster, not an 18-electron complex. Forcing the 18-electron rule here would invent excessive bond orders. chloride bridges donate to more than one center, so summing local counts is not the same as counting distinct electrons of the whole cluster.

Use an ${\eta }^5$ cyclopentadienyl ligand and four terminal carbonyl ligands at each titanium, with a Ti–Ti single metal-metal bond. The middle panel shows this connectivity. Neutral electron counting assigns five electrons to each Cp and givesEquivalently, the two Cp anions require an average oxidation state Ti(I), hence $d^3$; ionic counting gives $3+6+8+1=18$. The formal Ti–Ti bond order is one. The large coordination envelope can make this bond weak and long; reaching 18 electrons is bookkeeping, rather than proof of a short, strong bond. There is no need to invent extra CO bridges or a higher Ti–Ti bond order.

In the chromium complex, Cr(0) is $d^6$ and receives substantial donation from four carbonyl ligands. Its occupied orbitals can give pi backbonding into the boron-rich acceptor orbitals of the B${}_2$N${}_2$ ring, while the nitrogen-rich occupied ring orbitals donate to the metal. This favors binding across the ring face, approximately ${\eta }^4$, with all four ring atoms participating. A nearly planar ring maintains the p-orbital alignment needed for that interaction and accounts for comparable Cr–B and Cr–N distances. A four-electron ring donor also gives the useful count $6_d+8_{\mathrm{CO}}+4_{\mathrm{ring}}=18$.

Ti(IV) in TiCl${}_4$ is $d^0$ and strongly electron deficient. It is a good acceptor for the two nitrogen lone pairs, but has no filled $d$ set for comparable backbonding into the boron acceptors. chromium therefore favors a face-bound pi complex, while titanium favors N,N sigma coordination. Donation makes the nitrogen environments more nearly tetrahedral and permits the 2,4-Diazadiboretidine ring to pucker. The borons are not directly bound in the same way, explaining $d(\mathrm{Ti}\cdots \mathrm{B})>d(\mathrm{Ti}-\mathrm{N})$. The distinction follows from electron availability, rather than simply calling every short metal–ring contact a separate ordinary sigma bond.

The first equivalent of titanocene methylidene methylenates the ester: the carbonyl oxygen and titanium form an oxatitanacyclobutane, whose fragmentation gives $\mathrm{PhC}(\mathrm{OMe})=\mathrm{C}{\mathrm{H}}_2$ and a titanium–oxygen product. The second equivalent adds across this electron-rich double bond. The initial adduct is a titanacyclobutane with cyclic connectivity $\mathrm{C}{\mathrm{p}}_2\mathrm{Ti}-\mathrm{C}{\mathrm{H}}_2-\mathrm{C}(\mathrm{Ph})(\mathrm{OMe})-\mathrm{C}{\mathrm{H}}_2-$, closing from the last methylene to titanium. The ester's phenyl-substituted carbon lies opposite Ti; the two new methylenes originate from the two methylidene equivalents.

Beta-methoxy elimination opens that metallacycle to B, $\mathrm{C}{\mathrm{p}}_2\mathrm{Ti}(\mathrm{OMe})[\mathrm{C}{\mathrm{H}}_2\mathrm{C}(\mathrm{Ph})=\mathrm{C}{\mathrm{H}}_2]$, the allyltitanium reagent. The C–OMe bond breaks, OMe transfers to Ti, one Ti–C bond becomes the new C=C bond, and the other Ti–C bond remains as the allylic nucleophile. The cyclic adduct is therefore a precursor to the opened reagent B; treating B as merely the one-equivalent enol ether loses the second reagent carbon.

The aldehyde oxygen coordinates to titanium. Allylic carbon transfer to its carbonyl carbon, accompanied by movement of the C=O pi electrons to oxygen, gives the titanium alkoxide $\mathrm{PhC}{\mathrm{H}}_2\mathrm{C}{\mathrm{H}}_2\mathrm{CH}(\mathrm{O}-\mathrm{Ti})\mathrm{C}{\mathrm{H}}_2\mathrm{C}(\mathrm{Ph})=\mathrm{C}{\mathrm{H}}_2$. Acidic aqueous workup cleaves O–Ti and protonates oxygen, giving $\mathrm{PhC}{\mathrm{H}}_2\mathrm{C}{\mathrm{H}}_2\mathrm{CH}(\mathrm{OH})\mathrm{C}{\mathrm{H}}_2\mathrm{C}(\mathrm{Ph})=\mathrm{C}{\mathrm{H}}_2$. A six-membered allyl-transfer arrangement is a convenient mechanistic model. No enantioselectivity is imposed by these achiral reagents.

Both free primary allylic alcohols undergo Sharpless asymmetric epoxidation. With slightly more than two equivalents of hydroperoxide, the expected fully oxidized product is the diepoxide, with both original C=C bonds converted to epoxides and both primary alcohols retained. The two central pre-existing stereocentres and their ether substituents are retained.

The stereochemical construction is unambiguous by applying the L-(+)-tartrate mnemonic separately at each end: orient that alkene horizontally with its CH${}_2$OH group at lower right, and deliver oxygen from the back face. In the source's original in-plane orientation this gives the left epoxide oxygen behind the plane and the right epoxide oxygen in front. The methyl and hydroxymethyl groups retain their original relative arrangement about each former double bond. One cannot simply draw both oxygens on the same face of the original extended chain: the two allylic-alcohol ends have opposite drawing orientations. titanium binding to each free hydroxyl directs oxygen transfer; the pre-existing stereocentres can create matched/mismatched rate differences, but the stated reagent control predicts the stereochemistry of both epoxidations. Molecular sieves maintain a dry active titanium–tartrate system.^[ref]

ferroelectric poling is essential. An initially unpoled ceramic has differently oriented crystal grains and domains whose charge responses largely cancel. An applied electric field favours suitably oriented domains and leaves a remanent macroscopic polarization after removal; it does not rotate all crystal grains into a single crystal.

In the linear coefficient model the added source polarization isThis is the stress-generated contribution at fixed electric field, equivalent to the short-circuit charge per face area. Under open-circuit conditions the induced field supplies an opposing dielectric contribution, so $\Delta D=0$; the source quantity must not be confused with a change of total free electrode charge. For scale, room-temperature single-domain barium titanate has spontaneous polarization of order $0.26\mathrm{C}{\mathrm{m}}^{-2}$: the calculated increment is about $3.0\times 10^{-4}$ of that value, a small response rather than wholesale reversal of the spontaneous polarization.^[ref]

For the requested displacement estimate, assign the whole additional dipole moment of one cell to movement of its single $\mathrm{T}{\mathrm{i}}^{4+}$ ion relative to a fixed background. Then $\Delta P=(4e)\Delta x/(a^{2}c)$, givingThis formal-charge estimate is only an equivalent displacement: real polarization changes involve oxygen and barium displacements, electronic polarization and Born effective charges, as well as possible domain-wall motion. The coefficient supplied for the ceramic cannot uniquely determine an actual microscopic titanium trajectory.

For Ti–6Al–4V, use its own fracture toughness and yield stress inThe same unit-conversion formulas from part(i) apply. A feasible yield-limited screening interval requires $t_{\mathrm{yield}}<t_{\max }$, using consistent crack geometry and low-temperature properties.

High specific strength and corrosion resistance can be attractive, but cost, fabrication, joining and toughness must also be assessed. The label HCP is not a sufficient rule that every titanium alloy has the same sharp ductile-to-brittle transition as ferritic steel; actual low-temperature alloy and microstructural data are needed.

TODO: exact numerical thickness pending the 2018 NST Data Book's Ti–6Al–4V property pair. No historical value or material-specific Arctic toughness is invented.

Enantioselective oxygen transfer to an allylic alcohol using a titanium–tartrate complex and a hydroperoxide. With the alkene horizontal and its hydroxymethyl substituent at lower right, L-(+)-tartrate directs delivery from the back face and D-(−)-tartrate from the front face. Rotate a structure in its plane to apply the mnemonic; reflecting it changes its stereochemistry.

A four-membered metallacycle containing titanium and three carbon atoms. Cycloreversion can exchange an alkene and a titanium alkylidene; other eliminations become available with suitable substituents.