A dimensionless estimate of displacement events per target atom under a specified damage model. It does not count the vacancies still present after recombination and microstructural evolution.

The minimum recoil energy required, for a specified atom and direction or a stated averaging convention, to produce a surviving displacement defect. It depends on material, direction and the defect-survival criterion.

atomic displacements produced by collision-driven momentum transfer. It differs from the much larger electronic energy deposition that a fast alpha particle can produce along its track.

Annihilation of a vacancy and an interstitial, or related relaxation of irradiation defects. The surviving damage after a cascade can be much smaller than its initial number of atomic displacements.

Removal of an atom from its original lattice site. Whether it leaves a surviving defect depends on subsequent collisions, recombination and relaxation.

The dianionic, closed-shell O,O donor obtained by two deprotonations of catechol.

A free radical monoanion intermediate in redox state between a quinone and a catecholate; an isolated ligand has spin $S=1/2$. Its spin may couple to a metal spin in a complex.

A Cardano triplet is a triple of positive integers $(a,b,c)$ satisfyingwith real cube roots. Put $u=\sqrt[3]{a+b\sqrt{c}}$ and $v=\sqrt[3]{a-b\sqrt{c}}$. Cubing $u+v=1$ gives $uv=(1-2a)/3$, and cubing that product yields the necessary integer identityIt is also sufficient. From this identity, the real cube root of $a^2-b^{2}c$ gives $uv=(1-2a)/3$. Therefore $s=u+v$ satisfiesSince $a\ge 1$, the quadratic polynomial has negative discriminant $1-8a$ and no real roots. Hence $s=1$. This checks that cubing introduced no extraneous real solutions.

Reducing the integer identity modulo three forces $a\equiv 2(\mathrm{mod}3)$. Writing $a=3k-1$ for a positive integer $k$ then reduces it to $b^{2}c=k^2(8k-3)$. For example, $k=1$ gives $(a,b,c)=(2,1,5)$.

A primitive Diophantine solution is an integer tuple satisfying the equation whose coordinates have greatest common divisor one. For homogeneous equations, this removes common integer scale factors.

Cornacchia's algorithm searches for primitive integer solutions of $x^2+d{y}^2=m$ by combining a modular square root of $-d$ modulo $m$ with the Euclidean algorithm. In the standard odd-prime case $m=p$, choose a root $r$ with $0<r<p/2$, run the Euclidean algorithm on $(p,r)$ until a remainder $x<\sqrt{p}$ occurs, and test whether $(p-x^2)/d$ is a nonnegative integer square. Both roots can be tried.

For $p=13,d=1$, the root $r=5$ of $-1$ gives the remainder $3$, and $13-3^2=4$, yielding $(x,y)=(3,2)$. The final exact square test is essential: a modular root is only a candidate-producing input. For general composite moduli, additional hypotheses and choices are required to enumerate primitive solutions.

Diffusional material creep dominated by transport through grain interiors. For an equiaxed-grain model, $\dot{\epsilon }\sim C{D}_V\Omega \sigma /(k_{B}T{d}^2)$, with geometric factor $C$, atomic volume $\Omega$ and grain size $d$. It is linear in stress in this ideal limit.

Diffusional material creep dominated by grain-boundary transport. An ideal scaling is $\dot{\epsilon }\sim C\delta D_{GB}\Omega \sigma /(k_{B}T{d}^3)$. Its stronger grain-size dependence and typically lower transport activation energy distinguish it from bulk-diffusion material creep.

A para-tethered phenoxide can react through its carbon-nucleophilic resonance contributor. Intramolecular displacement of a tether leaving group at the para carbon creates a spiro ring and a cyclohexadienone. Subsequent acid-induced bond migration to a neighbouring carbon can expand the attached ring and restore the phenol. The two-stage sequence can therefore give a final substitution position different from the initial carbon attack.

Buta-1,3-diene is $\mathrm{C}{\mathrm{H}}_2=\mathrm{CH}-\mathrm{CH}=\mathrm{C}{\mathrm{H}}_2$, a four-carbon diene with a conjugated system. Adjacent p orbitals allow electron delocalization across the two double bonds and the intervening single bond.

A suitable cisoid conformation can react with a dienophile in a Diels-Alder reaction. The need for the reactive conformation connects internal rotation with cycloaddition reactivity.

A five-carbon cyclic conjugated diene. Cycloaddition across its two double bonds produces a norbornene framework while retaining the original five-membered carbon ring.

The unsaturated partner that combines with a conjugated diene in a Diels–Alder reaction. Electron-withdrawing substituents often enhance reactivity in a normal-electron-demand process.

A preference for the cycloaddition orientation in which relevant dienophile substituents lie under the diene framework in the transition geometry. It is distinct from regiochemistry and is not an unconditional guarantee of one product.

A reagent combination that installs the two saturated carbons of an ethylene cycloaddition while allowing more favourable reactivity. A removable electron-withdrawing substituent can activate the dienophile and be removed later.

A common kinetic preference for an orientation with dienophile substituents directed beneath the diene pi system. Steric and orbital interactions compete, so endo preference is not an absolute rule.