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An atmospheric thermal inversion has outward-increasing temperature. In a semi-grey irradiated atmosphere, define downward thermal optical depth $\tau$, constant opacity ratio $\gamma ={\kappa }_{\mathrm{vis}}/{\kappa }_{\mathrm{IR}}>0$, irradiation temperature $T_{\mathrm{irr}}$ and internal effective temperature of a planet $T_{\mathrm{int}}$. In the isotropic-irradiation closure, write the incident mean irradiance as $f_r{\sigma }_{\mathrm{SB}}{T}_{\mathrm{irr}}^4$, with $f_r>0$ explicitly specifying the averaging convention. Common choices are $1/4$ for a whole-sphere average and $1/2$ for a dayside average.

The analytic radiative equilibrium profile in this closure isDifferentiation givesSince $\tau$ increases inward, an atmospheric thermal inversion occurs when this derivative is negative. Thus $\gamma >1$ is necessary in this closure; a nonzero intrinsic radiative flux also requires sufficiently strong irradiation. For $T_{\mathrm{int}}>0$, any inversion is restricted toprovided the logarithm is positive. The $T_{\mathrm{int}}=0$ limit gives the threshold $\gamma >1$. These coefficients depend on the angular closure; a collimated beam has an incidence-angle dependence.^[ref]

This is a semi-grey irradiated atmosphere in radiative equilibrium. The quantities have these meanings:

The model assumes a static plane-parallel atmosphere, separate grey stellar and thermal bands with constant opacities, local thermodynamic equilibrium for thermal emission, negligible scattering, and transport by radiative transfer rather than convection. The Eddington closure approximation sets the thermal angular-moment ratio to $1/3$, and the upper boundary supplies the usual $2/3$ term. Treating the incoming radiation with representative direction cosine $1/\sqrt{3}$ gives its attenuation $e^{-\sqrt{3}\gamma \tau }$. Intrinsic flux enters from below, while the imposed external stellar flux is absorbed from above.

With constant gravity and hydrostatic equilibrium, $\tau \simeq {\kappa }_{\mathrm{IR}}P/g$, allowing conversion to an atmospheric pressure-temperature profile. This last relation additionally assumes constant thermal opacity. The grey treatment describes the energy balance approximately; it does not resolve individual molecular absorption lines.

The internal effective temperature of a planet $T_{\mathrm{int}}$ parametrizes intrinsic cooling, while the irradiation temperature $T_{\mathrm{irr}}$ parametrizes incident stellar flux before the redistribution factor $f$. The quantityis the visible-to-thermal mean-opacity ratio, and $\tau$ is downward thermal optical depth. Under the Eddington two-stream boundary condition,while the common semi-grey choice is $c=\sqrt{3}$.

For a young giant at $40\mathrm{au}$, stellar heating is weak and a still-large $T_{\mathrm{int}}$ dominates. Its pressure-temperature profile rises steadily inward, approximately as $T^4\propto b+\tau$, and joins a deep convective adiabat.

For a hot Jupiter close to a Sun-like star, $T_{\mathrm{irr}}\gg T_{\mathrm{int}}$. It has a broad, nearly isothermal irradiated radiative layer, followed by a deep rise where intrinsic flux and increasing opacity matter. If $\gamma >1$, absorption of starlight above the thermal photosphere can create an atmospheric thermal inversion.

For a temperate sub-Neptune around an M dwarf, irradiation and internal cooling can be more comparable. Its profile generally has a moderate radiative layer above a convective interior; near-infrared stellar radiation, molecular opacity, clouds, and hazes determine whether the upper profile is weakly inverted or decreases outward.

Neutron capture by boron-10 can generate helium, leading to helium embrittlement, especially at elevated temperature. The relevant nuclear reaction isThe emitted alpha particle eventually becomes a helium atom in the metal. The product branch can also involve gamma emission, without changing this helium-producing balance. The production depends on boron isotopic content and the neutron energy spectrum; a thermal component is particularly relevant to this reaction.

Helium has very low solubility in steel and can be trapped at vacancies, interfaces and grain boundaries. Boron already segregated near grain boundaries can therefore create a locally unfavorable distribution of helium. Stabilized helium–vacancy clusters develop into bubbles or cavities. Under tensile stress, boundary cavities can grow and link, reducing ductility, fracture toughness and creep-rupture life through intergranular fracture. Early thin-foil observations and mechanical tests associated high-temperature ductility loss with helium-containing boundary bubbles and their coalescence.^[ref]

Irradiation temperature controls the transport and clustering of both helium and vacancies. At lower temperature, reduced long-range transport can retain small trapped clusters in the matrix. These can contribute to irradiation hardening, but extensive boundary-cavity growth is kinetically less favorable. At higher temperature, migration, cavity growth and material creep become faster, making intergranular fracture and loss of high-temperature ductility more likely. Trap distribution and prior microstructure can redistribute the gas, so there is no composition-independent critical temperature or universal monotonic law.

The mechanical-test temperature also matters. Material irradiated cold can become embrittled during subsequent hot testing or service when trapped gas and defects become mobile. Distinguish gas production during irradiation from the later conditions under which its mechanical consequences are expressed.

Void swelling is generally favored by an intermediate irradiation temperature window. The balance involves defect mobility, stable nucleation, absorption at defect sinks and thermal emission; a larger vacancy population alone does not guarantee a larger void.

At low temperature, vacancy migration is slow. Self-interstitials may migrate and form dislocation loops, and small clusters can cause considerable irradiation hardening, while vacancies cannot readily feed large growing cavities. Defects retained in the matrix can evolve during later heating.

At intermediate temperature, vacancies can migrate far enough to reach stable cavities. Interstitial bias and the competition among defect sinks sustain a net vacancy flux into these cavities. Once the incubation period has produced suitable nuclei and sink populations, substantial void swelling can develop.

At high temperature, the thermal equilibrium vacancy population and vacancy emission from small cavities rise. A small void's curvature further favors emission, so maintaining stable nuclei becomes harder. Recovery changes the dislocation population, and recombination, gas transport and evolving precipitates also change the flux balance. The irradiation-driven vacancy supersaturation can fall, suppressing void nucleation and net swelling. Existing cavities may still coarsen, and gas bubbles, material creep and other damage can remain important.

Thus low-temperature defect retention, intermediate-temperature cavity growth and high-temperature recovery/emission are different regimes. The boundaries depend on alloy composition, initial microstructure, gas production, accumulated displacements per atom and displacement dose rate. No alloy-independent peak temperature can be inferred from the paper's data.

Use the same atomic fractions and collision assumptions as in part(c). Let $A$ be the irradiated area. The number of atoms in the sheet is $N_t=ntA$, while the number of incident neutrons is $\Phi A$, where $\Phi$ is the neutron fluence.

Counting one primary knock-on atom for each neutron that scatters and using $\nu =430$ displacements per recoil gives the first-collision estimateHence the nominal damage is about $55$ displacements per atom under the assumed composition. In the thin-sheet approximation the density and thickness cancel:The $1.4\%$ difference comes from using first-collision attenuation rather than the linear approximation. Treating repeated collisions after deflection would require angular and energy information not provided; this correction is already smaller than the precision warranted by the composition assumption.

There is also a counting convention in “other atoms”: $430$ counts secondary displacements, while including the displaced primary knock-on atom itself gives $\nu =431$. The corresponding first-collision result is $54.58$ rather than $54.45$ dpa, an increase of only $0.23\%$. Both conventions round to about $55$ dpa. The numerical estimate is explicitly conditional on assigning the stated displacement yield to each scattering event.

A value greater than one is physically possible. Displacements per atom counts displacement events relative to the initial atom inventory: atoms can be displaced repeatedly, and many vacancies and self-interstitials recombine. It does not mean $55$ surviving vacant sites per original lattice site, or that every atom has moved exactly $55$ times. This is a large nominal exposure for which evolving dislocation loops, cavities and irradiation hardening matter; the surviving microstructure additionally depends on irradiation temperature, displacement dose rate, recombination and defect sinks.

A semi-grey irradiated atmosphere uses separate averaged opacities for incoming stellar light and outgoing planetary thermal radiation. Their ratio $\gamma ={\kappa }_{\mathrm{vis}}/{\kappa }_{\mathrm{IR}}$ compares where the two energy streams interact with matter. Large visible opacity relative to infrared opacity deposits stellar heat at a relatively small thermal optical depth; sufficiently strong high-altitude heating can produce an atmospheric thermal inversion.^[ref]

The model connects irradiation temperature, internal effective temperature of a planet and heating depth to an atmospheric pressure-temperature profile. It is useful for interpreting broad thermal structure, but averaging each band suppresses spectral windows and molecular lines that matter in detailed atmospheric retrieval.

This factor converts substellar incident flux into the representative flux used in an atmospheric model. Uniform global averaging gives $f_r=1/4$; dayside averaging gives $1/2$ if albedo is treated separately. The convention must be stated when defining an irradiation temperature.