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In selective-area epitaxy, precursors supplied by metal-organic vapour-phase epitaxy incorporate on exposed crystalline InP through patterned openings in a silicon-nitride mask. Epitaxial facet-dependent incorporation and surface diffusion select axial and radial growth rates. Advantages include lithographic position and pitch control, an epitaxial orientation relationship and the absence of a foreign metal catalyst. Limitations include mask patterning, selective nucleation requirements, facet-dependent composition, loading and pitch effects, and unwanted radial growth.

In vapour-liquid-solid growth, precursors decompose and supply material to a liquid catalyst droplet; supersaturation drives precipitation at the liquid-solid interface. Droplet size can control wire diameter and enable high aspect ratios. Limitations include possible catalyst contamination, droplet stability and coarsening, composition memory at heterointerfaces, and restrictions from the catalyst phase diagram. Patterned catalyst placement can also control position, so positional ordering is not exclusive to selective-area growth. Conversely, selective-area wires are not automatically free of taper or compositional interfaces.^[ref]

Start with a lightly boron-doped p-type Si wafer. Define an implantation opening for the pMOS region using an oxide/nitride implantation mask, then implant donor ions, commonly phosphorus, at energies and doses giving the required well depth and concentration. An anneal activates donors and repairs damage; thermal drive-in can deepen and smooth the well profile. The donor concentration must exceed the original acceptor concentration in the well. Retain p-type material outside it for the nMOS device. Form a later n+ well contact to tie the n-well to the positive supply; the p+ substrate contact ties the substrate to ground. Diffusion from a donor source is an alternative to ion implantation and drive-in.

Use two-dimensional NOESY, or ROESY where molecular tumbling makes NOESY less favorable, to look for through-space correlations between proton sites on the two ends that are brought together in the closed conformation. Assign resonances first and compare predicted ring contacts with an open reference or model.

The nuclear Overhauser effect involves NMR cross-relaxation through magnetic dipole–dipole coupling. Its strong approximate $r^{-6}$ dependence makes it sensitive to short interproton distances. A NOESY cross-peak therefore reports spatial proximity, including between sites far apart along the covalent skeleton; it is not a NMR scalar coupling transmitted along that skeleton.

Run a dilution series: end-to-end contacts that persist at concentrations where intermolecular association is negligible support intramolecular ring closure. Hydrogen-bonded NH chemical shifts and diffusion measurements can support the assignment, but neither alone proves ring topology. A flexible open chain can transiently bring ends together, and intermolecular contacts can also give NOEs. Several geometry-consistent cross-peaks, concentration behavior and controlled mixing times make the case much stronger than one peak.

Use a shaped nickel particle or nanowire as both sacrificial template and reducing agent. gold grows where nickel is consumed, allowing gold-coated particles, porous shells and, with suitable removal of the remaining nickel, hollow gold nanostructures. The original template controls the overall shape; the balance of dissolution, deposition and diffusion controls wall thickness and porosity.

A hollow product is not guaranteed simply by writing the reaction: deposition may seal a solid nickel core, whereas defects can maintain access to the interior and permit further replacement. Controlled composition and transport can give a progression from core–shell to porous or hollow structures. This is useful for high-surface-area electrodes, catalytic surfaces and plasmonic structures, while avoiding an electrical lead to every individual nanoparticle.

Clean and degrease the nickel surface, remove its inhibiting oxide by an appropriate brief surface treatment, rinse, and immerse it in an aqueous bath containing reducible gold(III) species. Control concentration, temperature and immersion time, then rinse after the desired deposition. A solution of a gold complex is practical, but its actual reduction potential must be evaluated for that complex and bath: it need not equal the supplied free-ion standard potential.

At anodic sites nickel dissolves as $\mathrm{N}{\mathrm{i}}^{2+}$, releasing electrons into the conducting substrate. Those electrons reduce gold ions at cathodic sites, nucleating and growing metallic gold. The nickel itself is the reducing agent, so no external electrical supply or separate reducing reagent is required. The coupled electron balance is three nickel atoms dissolved for two gold atoms deposited for the stated oxidation states.

The process can slow or stop when a sufficiently continuous gold coating prevents the electrolyte from reaching nickel. This is not an absolute thickness rule: pores, cracks or exposed edges preserve nickel dissolution and allow continued deposition, sometimes with undercutting. nickel passivation, depletion of gold ions near the surface, diffusion through a porous layer, bath speciation and competing reactions also affect the rate. Dense gold coverage and porous gold coverage therefore have very different limiting behavior.

forward bias opposes the built-in electric field and lowers the junction barrier from approximately $V_{\mathrm{bi}}$ to $V_{\mathrm{bi}}-V$. In the abrupt-junction depletion approximation,Thus the depletion region narrows and carriers cross the reduced barrier more readily. electron injection into the p side and hole injection into the n side increase strongly, allowing radiative recombination in an LED.

The injected minority-carrier concentrations, and therefore the ideal diffusion electric current, vary approximately as $\exp (qV/k_{B}T)$. This produces the steep forward branch of the electric current–voltage curve. One should not extrapolate the depletion-width formula to claim that the junction must lose its entire depletion region before conducting; appreciable electric current flows while a barrier remains, and high injection and series resistance eventually invalidate the simple formulas.

In the imposed free-draining polymer friction model, the drag coefficients of the $N$ monomers add, giving ${\zeta }_1=N(6\pi \eta a)$ and $D_1=k_{B}T/(6\pi \eta aN)$. Taking the same three-dimensional centre-of-mass RMS convention as part (ii),The polymer is a one-dimensional connected contour, but that alone does not restrict its centre of mass to one spatial axis. If the intended coordinate is instead literal one-dimensional curvilinear motion along an entanglement tube, its RMS contour displacement is $\sqrt{2D_{1}t}=\sqrt{k_{B}Tt/(3\pi \eta aN)}$. That is a different coordinate and should not be compared with a three-dimensional displacement without stating the conversion/model.

Additive local drag is the free-draining Rouse model approximation. It neglects long-range solvent-mediated hydrodynamic interactions and is not a complete theory of an entangled chain. In a tube/reptation description, curvilinear diffusion can scale as $N^{-1}$ while long-time spatial centre-of-mass diffusion scales as $N^{-2}$ after the chain renews its tube. Conversely a dilute unentangled coil can exhibit Zimm model hydrodynamic coupling. Monomer motion, centre-of-mass motion and contour sliding therefore cannot all be identified with the same displacement law.

For a no-slip spherical particle in a Newtonian fluid at low Reynolds number, Stokes law gives $\zeta =6\pi \eta a$. The fluctuation-dissipation theorem then gives the Stokes–Einstein relation $D=k_{B}T/\zeta$. Using the conventional radial Brownian displacement in three dimensions,Here $k_B$ is the Boltzmann constant, $T$ absolute temperature, $\eta$ dynamic viscosity, $a$ particle radius and $t$ elapsed time. Larger thermal forcing raises diffusion; larger viscous drag reduces it. The formula requires times long relative to momentum relaxation and neglects walls, interactions and external drift.

The word “average” needs a statistic: the vector mean is zero, while the mean radial displacement for Gaussian three-dimensional diffusion is $⟨|\Delta \mathbf{r}|⟩=4\sqrt{Dt/\pi }$, not $\sqrt{6Dt}$. Both scale as $t^{1/2}$. Neither is the total length of a microscopic random path.

The two historically dominant commercial processes are gaseous diffusion enrichment and gas centrifuge enrichment. The first exploits differences in molecular transport through a barrier; the second exploits mass-dependent redistribution in a rotating gas. Both are isotope-separation principles rather than chemical separation of uranium from another element.

gas centrifuge enrichment had become predominant by the time of this examination. Its much smaller electricity demand reduces a major operating cost of diffusion technology. Modular capacity and the replacement of ageing diffusion plants contributed to commercial adoption. This historical conclusion does not require machine dimensions, operating conditions or a cascade design. The United States' final operating diffusion plant closed in 2013, consistent with the earlier transition rather than a claim based on current plant counts.^[ref]

The key imbalance is preferential loss of self-interstitials to biased defect sinks, leaving a vacancy supersaturation that can feed void swelling. The initial displacement cascade creates both vacancies and self-interstitials, often in equal net numbers as Frenkel pairs; it need not create more vacancies than interstitials at birth.

Many nearby Frenkel pairs recombine. Others separate, become trapped or form dislocation loops. Surviving self-interstitials are usually much more mobile than vacancies over the relevant range. Dislocations also have an interstitial bias: their elastic interaction with defects makes them more efficient sinks for self-interstitials. Preferential interstitial removal to the dislocation structure can leave the mobile vacancy population above its thermal equilibrium value.

A minimal radiation damage rate theory makes the distinction explicit:Here $c_i,c_v$ are defect number densities, $G$ is the surviving mobile-pair production rate per volume, $K$ the recombination coefficient, $D_i,D_v$ the diffusion coefficients, and $k_i^2,k_v^2$ the respective sink strengths. The vacancy loss term includes thermal emission in an effective form. Equal production terms do not require equal steady populations or equal partitioning among sinks. This simplified model omits an explicit cluster-size distribution.

A stable cavity receiving a greater vacancy influx than interstitial influx and vacancy emission grows. Vacancies therefore condense into voids, producing void swelling. Helium can stabilize small cavity nuclei, but a growing vacancy void need not be a high-pressure gas bubble. Dislocation loops, precipitates and changes in the dislocation density evolve alongside the voids and modify the competing sink strengths.

There is no violation of atom conservation: atoms represented by removed self-interstitials have joined extended defects or surfaces, while the corresponding unfilled lattice volume can accumulate in cavities. The classic electron-microscope observations of cavities in irradiated type316 steel provide direct evidence for this microscopic basis of swelling.^[ref]

For the prescribed homogeneous one-group model, number balance isUse steady state, $\partial n/\partial t=0$, and the diffusion approximation $\mathbf{J}=-D\nabla \varphi$. A spatially constant $D$ givesFor a spherically symmetric scalar field,Dividing by $D$ reproduces the requested radial equation. The assumptions are a steady source, spherical symmetry, homogeneous effective coefficients, a single energy-group description and a diffusion closure. In particular, replacing the general production term by the supplied $(\eta -1){\Sigma }_a\varphi$ is a model assumption, not a general identity for heterogeneous reactor materials.

Temperature controls the competition between defect creation, diffusion, recombination, sink absorption and thermal vacancy emission. Consequently void swelling is normally largest in an intermediate temperature window, whose location depends on alloy, dose rate and exposure history.

At low temperature, many generated defects lack sufficient mobility to feed a growing cavity. Considerable irradiation hardening can still occur through small defect clusters, while large-scale void growth remains limited.

At intermediate temperature, vacancies and interstitials migrate sufficiently to establish the biased defect fluxes described above. Cavity nuclei can survive and receive a net vacancy flux, producing strong swelling.

At sufficiently high temperature, thermal vacancy emission, recombination, recovery and changes in sink populations compete more strongly with vacancy absorption. Stable nucleation and net cavity growth can become less favorable. This does not imply that every radiation effect disappears: material creep, precipitation, segregation and gas-bubble evolution can remain important.

For austenitic stainless steels, the relevant range is typically several hundred degrees Celsius; an alloy-independent peak temperature is not justified. Raising the displacement dose rate changes the balance between production and thermally activated processes and can shift both the temperature window and the incubation dose. Temperatures from accelerated ion or electron experiments should therefore be interpreted together with dose rate, implantation, specimen thickness and surface-sink effects.

The resulting dislocation density reflects competing production, growth and recovery rates. The irradiation spectrum and displacement dose rate set the supply of recoil cascades and surviving point defects. Cascade structure controls the fraction born as isolated defects or clusters; equal nominal displacements per atom can therefore give different defect populations.

Temperature sets vacancy and interstitial mobilities and the rates of thermal emission and recovery. A simple radiation damage rate theory model illustrates the competition using number densities $c_i,c_v$:Here $G$ is the surviving mobile-pair production rate per volume, $K_{\mathrm{rec}}$ is the recombination coefficient, $D_i,D_v$ are diffusion coefficients, and $k_i^2,k_v^2$ are sink strengths. The model omits explicit cluster-size populations and thermal-equilibrium emission; those must be added for a quantitative description of a particular exposure.

The initial dislocation density, grain size, free surfaces, precipitates and existing cavities all change sink strengths. Dislocations can have an interstitial bias, so their capture rates for the two defect types differ. Solutes and transmutation gases alter defect binding and mobility. Crystal structure, elastic anisotropy and applied stress affect which loop orientations survive and grow.

For a number density $N_{\ell }$ of circular loops of radius $R$, their contribution to line length per volume isThus loop nucleation and loop growth both increase dislocation density, while shrinkage, annihilation and absorption at other sinks can decrease it. A change in loop count alone does not determine the change in total line density.

Schematically, ${\dot{\rho }}_d$ is a production-and-growth contribution minus recovery and annihilation. An initial rise followed by approximate saturation is possible when these balance, but the saturation value and approach rate depend on the evolving microstructure. There is no universal dislocation density fixed by dose alone.

Mixed oxide fuel begins as a specified mixture of uranium and plutonium oxides. A fuel pin develops strong radial temperature gradients during operation. The components and oxygen need not remain spatially uniform: their chemical potentials depend on temperature, local composition and oxygen-to-metal ratio.

Actinide redistribution in oxide fuel can involve thermally driven solid-state diffusion and evaporation–condensation transport across pores or a central cavity. Oxygen redistributes as well, changing local oxidation states and the relative thermodynamic tendencies of uranium- and plutonium-bearing species. Pore migration, grain growth and central-void formation couple compositional redistribution to fuel restructuring. Differential local burnup also changes composition over time.

The direction and magnitude of plutonium migration cannot be inferred from a temperature gradient alone. In particular irradiated fast-reactor pins, enrichment around a central cavity has been observed, but the result depends on stoichiometry and operating history. M. Bober and colleagues investigated thermal-diffusion and evaporation–condensation effects using autoradiographs of sections of fuel irradiated in RAPSODIE.^[ref]

Redistribution changes the radial heat generation rate and material properties. Concentrating more fissile material near an already hot region can raise local heat generation; changes in thermal conductivity and local melting behavior modify the temperature response. Gas release, pore formation and fuel–cladding contact can also change thermal resistance and mechanical loading. A spatially averaged composition can therefore underestimate a local temperature or chemical interaction.

Segregated fission products add another concern: mobile species can accumulate at colder interfaces or react with the cladding, while insoluble products form separate phases. Fuel performance must therefore track composition, temperature and microstructure together, rather than treating segregation as merely a cosmetic change in an otherwise fixed fuel.

A fast charged particle loses energy by ionizing and exciting silicon, creating electron–hole pairs along its path. In a reverse-biased microstrip diode, the electric field separates and drifts the electrons and holes toward opposite electrodes. Their motion induces a current on the strips, which is integrated by the readout electronics to locate the hit.

The bias depletes the bulk of mobile equilibrium carriers, reducing conduction and capacitance, and supplies rapid directed charge collection before recombination or trapping. Without a collecting field, diffusion is slow and charge spreads or recombines, degrading signal and position measurement. The signal is generated by the traversing particle's energy deposition, not by the bias creating the ionization charge.

Joining p-type and n-type regions initially allows majority carriers to diffuse across and recombine. Exposed negative acceptor charge on the p side and positive donor charge on the n side form a depletion region. Its electric field points from n to p and balances diffusion at equilibrium. The Fermi level is then spatially constant, while the conduction and valence band edges bend across the built-in potential; this equilibrium barrier cannot supply continuous power to a load.

forward bias makes p positive relative to n, reducing the barrier from $q{V}_{bi}$ to $q(V_{bi}-V)$. electrons enter p material and holes enter n material, becoming minority carriers whose diffusion and recombination sustain the current. reverse bias enlarges the barrier and leaves a much smaller minority-carrier current until breakdown. The ideal diffusion model gives $I=I_0[e^{qV/k_{B}T}-1]$, the Shockley diode equation; depletion-region recombination, series resistance and breakdown limit this approximation. Minority-carrier injection on both sides is essential to the junction theory.^[ref]

Absorption of photons with $h\nu \gtrsim E_g$ creates electron-hole pairs, the upward band-diagram arrow. In the depletion region the built-in field drives electrons towards n material and holes towards p material; diffusion also delivers carriers generated within a diffusion length outside that region. Selective contacts collect the two signs. Illumination maintains a nonequilibrium quasi-Fermi-level splitting, allowing a voltage and useful current through an external load; it is the light, rather than the equilibrium barrier, that supplies the energy.

Taking forward diode current as positive, an ideal illuminated junction haspower is delivered where $VI<0$ in this convention. Reflection, below-gap transmission, carrier thermalization, recombination and resistive losses limit real conversion; even an ideal single junction has radiative detailed-balance limits.^[ref]

Donor doping supplies conduction electrons to n-type semiconductor; acceptor doping supplies holes to p-type semiconductor. In the nondegenerate, fully ionized limit, majority densities are approximately $n_n=N_D$ and $p_p=N_A$, while minority densities follow $np=n_i^2$. On contact, majority carriers diffuse across the junction, exposing positive donor ions on the n side and negative acceptor ions on the p side. The depletion-region electric field points from n to p and opposes further diffusion. At thermal equilibrium the Fermi level is uniform and drift balances diffusion, with no net current.

For an abrupt junction with uniform doping and permittivity ${\epsilon }_s$, charge neutrality gives $N_D{x}_n=N_A{x}_p$. Poisson's equation then givesPositive applied $V$ means forward bias: it lowers the barrier and narrows the depletion region. Most depletion width lies in the more lightly doped side. The formula assumes the depletion approximation and fails once its operating assumptions break down.

forward bias increases minority densities at the depletion edges by the factor $e^{qV/k_{B}T}$. In the neutral material, the steady diffusion–recombination equation gives exponentially decaying excess minorities, with lengths $L_n=\sqrt{D_n{\tau }_n}$ and $L_p=\sqrt{D_p{\tau }_p}$. Taking the carrier gradients at the junction yieldsThis ideal diode law assumes low injection, nondegenerate statistics, negligible depletion-region recombination and series resistance, and sufficiently long neutral regions. Its exponential comes from the bias-dependent injection barrier; there is no universal abrupt “turn-on” voltage. reverse bias suppresses injection, giving an approximate $-I_S$ current until additional leakage or breakdown matters.^[ref]

Rectification uses the asymmetric current-voltage characteristic; controlled reverse breakdown enables voltage references, with current limiting. Bias-dependent depletion capacitance enables varactors. photodiodes and solar cells collect optically generated carriers through the junction field and diffusion toward it; efficient light-emitting diodes use radiative electron–hole recombination, preferably in a direct-gap material. Series resistance, high injection, trap-assisted recombination, heating and finite carrier storage explain departures from the ideal steady-state law.

A localized quantum particle is described by a wave packet: a coherent quantum superposition of momentum components, rather than a single delocalized plane wave. The Born rule assigns the position probability density $|\psi (x,t){|}^2$. A useful normalized initial Gaussian wave packet isIts mean position and momentum are $x_0$ and $p_0$, and its standard deviations satisfy ${\sigma }_x(0)={\sigma }_0$, ${\sigma }_p=\hslash /(2{\sigma }_0)$. This unchirped initial state saturates the Heisenberg uncertainty relation.

For a free particle, the momentum-space Schrödinger equation multiplies each amplitude by $e^{-i{p}^{2}t/(2m\hslash )}$. The inverse Fourier transform is a Gaussian integral, giving the free Gaussian wave packet evolutionOne way to obtain this expression explicitly is to write the initial momentum amplitude as proportional to $\exp [-{\sigma }_0^2(p-p_0{)}^2/{\hslash }^2-ip{x}_0/\hslash ]$, multiply by the free evolution phase and complete the square in $p$. Taking the modulus squared removes the position-dependent phase and leavesThe expectation value follows uniform classical motion while the density spreads. A narrower initial packet needs a wider momentum distribution and therefore spreads more rapidly. At long times its width grows linearly with $|t|$; its variance grows quadratically, unlike the linear variance growth of ordinary diffusion. The probability continuity equation preserves total probability despite the decreasing peak height. The mean momentum and the mean energy $(p_0^2+{\sigma }_p^2)/(2m)$ remain constant, since the momentum populations do not change.

The same result has a wave interpretation. For a free nonrelativistic particle $\omega (k)=\hslash k^2/(2m)$, so the group velocity is $d\omega /dk=\hslash k/m$, equal to $p_0/m$ at the packet center. The curvature $d^2\omega /d{k}^2=\hslash /m$ makes different momentum components travel with different group velocities, producing dispersion. The phase velocity is $\omega /k=\hslash k/(2m)$ for this energy convention and does not describe the motion of the envelope. The evolving quadratic spatial phase is essential to the spreading; a density snapshot alone does not specify the full quantum state.

A freely spreading Gaussian wave packet remains a pure Gaussian state but develops position-momentum correlations. Its symmetrized covariance is $C_{xp}=⟨\{x-⟨x⟩,p-⟨p⟩\}⟩/2=t{\sigma }_p^2/m$. ConsequentlyEquality in the Robertson-Schrödinger uncertainty relation survives at every time, whereas the simpler product ${\sigma }_x{\sigma }_p$ equals $\hslash /2$ only initially. A chirped Gaussian wave packet can already have $C_{xp}(0)\ne 0$. More generally, $x(t)=x(0)+p(0)t/m$ gives ${\sigma }_x(t{)}^2={\sigma }_x(0{)}^2+2t{C}_{xp}(0)/m+t^2{\sigma }_p(0{)}^2/m^2$. A sufficiently negative initial covariance permits initial contraction before subsequent spreading.

A potential energy changes this behavior. A quadratic potential preserves the Gaussian form because the evolution of position and momentum is linear. For a quantum harmonic oscillator, the Gaussian covariance in a harmonic potential isA coherent state has ${\sigma }_x^2=\hslash /(2m\omega )$, ${\sigma }_p^2=m\hslash \omega /2$ and zero centered covariance. Its width is constant, and its center follows $x_c(t)=x_0\cos \omega t+p_0\sin \omega t/(m\omega )$. In oscillator notation its coherent state label rotates as $\alpha (t)=\alpha (0)e^{-i\omega t}$, apart from the common ground-energy phase of the state vector. A squeezed coherent state generally has a width that breathes at $2\omega$. Thus being Gaussian does not imply either free spreading or constant width; the Hamiltonian operator and initial covariance determine the result.

For a general nonquadratic potential, different parts of the packet experience different nonlinear forces, and an initially Gaussian profile need not remain Gaussian. Ehrenfest theorem gives the exact relations $d⟨x⟩/dt=⟨p⟩/m$ and $d⟨p⟩/dt=-⟨V^{\prime }(x)⟩$. Replacing the last mean by $-V^{\prime }(⟨x⟩)$ is exact for a quadratic potential and approximate for a packet narrow compared with the scale on which the force varies. This connects packet motion to the correspondence principle without treating a quantum state as a definite classical path. Scattering can split a packet into reflected and transmitted components, while confinement can produce oscillations and interference between discrete energy eigenstates. Their relative phases evolve even when their populations remain constant. Localization, spreading and coherent motion all follow from the same linear unitary time evolution.

Take a p-n junction made from one semiconductor, with the p-type semiconductor on the left and the n-type semiconductor on the right. At thermal equilibrium the chemical potential $\mu =E_F$ is horizontal throughout. Before contact the chemical potential lies nearer the valence band on the p side and nearer the conduction band on the n side; after contact the band edges bend to permit one common value.

In this orientation both the conduction band edge $E_C$ and the valence band edge $E_V$ decrease from p to n by $e{V}_{\mathrm{bi}}$. For a homojunction their separation $E_g=E_C-E_V$ is constant. The acceptor impurity level $E_A$ lies just above $E_V$ where acceptors are present, and the donor impurity level $E_D$ lies just below $E_C$ where donors are present. These impurity levels follow the local band bending; they do not form one additional horizontal band across the device.

The shaded depletion region has few mobile electrons or electron holes and contains fixed negative ionized acceptor impurities on the p side and positive ionized donor impurities on the n side. Thus its electric field points from n to p. Its neutral outer regions have negligible electric field, so the band edges become flat there. In the depletion approximation, charge neutrality requires $N_A{x}_p=N_D{x}_n$; the symmetric drawing illustrates $N_A=N_D$ rather than assuming that equality is universal. Diffusion and field-driven drift current cancel at equilibrium: no applied voltage does not mean zero internal electric field or zero built-in electric potential difference.