Stephane Gin investigates the reaction of glass with water and the mechanisms controlling its long-term corrosion. His work on borosilicate glasses examines why reaction can slow to a residual rate as the material and surrounding solution evolve.^[ref]

The problem couples chemical equilibrium, transport and the structure of alteration products. A low measured release rate does not by itself identify which process limits reaction, making controlled solution conditions and characterization of the reacting material essential.

For $D,\gamma ,a>0$, apply the superposition principle to two translated solutions for Gaussian initial data for the heat equation. The displayed solution of $u_t=D{u}_{xx}$ has odd initial profile $A(e^{-\gamma (x-a{)}^2}-e^{-\gamma (x+a{)}^2})$. Its total spatial integral is zero. For $A>0$, it is positive on $x>0$ and negative on $x<0$ at every $t\ge 0$.

Writing it as $2A{s}^{-1/2}{e}^{-\gamma (x^2+a^2)/s}\sinh (2\gamma ax/s)$ shows how cancellation affects its late-time shape. On the diffusive scale $x=O(\sqrt{s})$, its leading form is $4A\gamma ax{s}^{-3/2}{e}^{-\gamma x^2/s}$. Thus the two lobe maxima asymptotically lie at $x=\pm \sqrt{s/(2\gamma )}$, while their absolute heights decay as $s^{-1}$. Their centers are not fixed at the initial Gaussian centers. The first spatial moment remains ${\int }_{\mathbb{R}}xu(x,t)dx=2Aa\sqrt{\pi /\gamma }$, as follows directly from the Gaussian integral and translation.

With $X_{12}=x+iy$, a spinor whose components have phases $e^{-i\varphi /2}$ and $e^{i\varphi /2}$ has $x+iy\propto e^{-i\varphi }$, so its physical azimuth is $-\varphi$. The usual positive azimuth for those phases uses $X_{12}=x-iy$. Both conventions are valid if used consistently.

An anti-Hermitian generator $S$ is chosen so $[S,H_0]=-H_{e\text{-}ph}$. Expanding $e^{S}H{e}^{-S}$ leaves $H_0+[S,H_{e\text{-}ph}]/2$ to second order. Eliminating virtual phonons produces an effective electron-electron interaction, plus energy shifts. Resonant denominators and higher orders limit this perturbative elimination.

After delay tracking and fringe stopping at a phase center, an off-axis direction retains delay of order $B\theta /c$ and rotation-induced fringe-rate offset of order ${\Omega }_{E}B\theta /\lambda$. Averaging across a wide channel or long integration then decorrelates that direction. The displayed inequalities give order-of-magnitude small-smearing conditions; exact coefficients depend on baseline geometry, averaging window and allowed loss. A larger field or longer baseline therefore demands finer frequency and time sampling.

For the radial induced-charge screening kernel, integrate the sine transform twice by parts, with $g(0)=g(\infty )=0$ and vanishing remaining endpoint terms. This gives ${\rho }_{\mathrm{ind}}=Ze\int g^{\prime \prime }(q)\sin (qr)dq/r^3$. A Cauchy principal value singularity at $Q>0$ contributes $C\pi \cos (Qr)$ to that integral: set $s=q-Q$ and use ${\int }_{-\infty }^{\infty }\sin (sr)ds/s=\pi$ for $r>0$. Smoothly localizing around $Q$ justifies the large-$r$ extension of the limits. In a three-dimensional isotropic Fermi gas, $Q=2k_F$.

A single isotropic three-dimensional valley with two equal spin populations has $n=k_F^3/(3{\pi }^2)$. A density oscillation at wavevector $2k_F$ therefore has the wavelength in the title. This depends on carrier density, independently of effective mass and dielectric screening strength. Extra valley degeneracy, spin polarization or reduced dimensionality changes the state-counting relation.

For constant self-inductance L, the steady induced current satisfies $L\dot{I}+RI=BA\omega \sin \omega t$, giving amplitude $I_0=BA\omega /\sqrt{R^2+{\omega }^2{L}^2}$ and phase lag. Multiplication by the rotating loop normal again gives a static moment plus a twice-frequency rotating moment of magnitude $A{I}_0/2$, independently of lag. Therefore replace the resistance-only denominator $R^2$ in the radiation power by $R^2+{\omega }^2{L}^2$. A wire resistance alone is not sufficient for an exact prediction if inductance is appreciable.

Linearization around a pumped background couples the perturbation at positive frequency offset to the conjugate of the negative-offset perturbation. Their matrix eigenvalues determine amplification; assuming two unrelated scalar phase shifts misses the instability.

The wavelength associated with a specified Fermi wavevector. A local-density response normally requires variation on a scale large compared with $k_F^{-1}$; numerical factors such as $2\pi$ must be kept explicit when comparing a variation scale with the full wavelength.

The convention $F_j(\eta )=\Gamma (j+1{)}^{-1}{\int }_0^{\infty }{x}^j/[1+e^{x-\eta }]dx$ is useful for parabolic-band carrier densities. In three dimensions, a spin-degenerate single valley has $n=2[m{k}_{B}T/(2\pi {\hslash }^2){]}^{3/2}{F}_{1/2}[(\mu -E_c)/(k_{B}T)]$. Other conventions omit the gamma normalization, so prefactors must be stated.

A population of excitons can approach a thermal distribution only if redistribution and cooling occur before too many excitons disappear. A useful necessary scale comparison is ${\tau }_{\mathrm{therm}}\ll {\tau }_{\mathrm{life}}$, where the lifetime includes radiative and nonradiative recombination and the thermalization time describes the relevant energy redistribution.

The inequality is not sufficient to establish exciton condensation: density, disorder, pumping and the stability of the excitons also matter. A continuously pumped short-lived population can have a stationary distribution without satisfying equilibrium statistical mechanics. Spectral narrowing or a nonlinear emission threshold can therefore support several interpretations. Relating such observations to condensation requires a model of creation, cooling, interactions and losses, as emphasized in exciton condensation evidence and competing optical signatures.

In elliptic cylindrical coordinates, write a vector field as $A=A_u{e}_u+A_v{e}_v+A_z{e}_z$ using the local orthonormal unit vectors. The scale factors of orthogonal coordinates are $(h,h,1)$, so the divergence and curl areApplying the divergence to the gradient of a scalar function gives $\Delta \varphi =({\varphi }_{uu}+{\varphi }_{vv})/h^2+{\varphi }_{zz}$. These formulas hold where $h>0$. They use physical components along unit vectors, rather than components along the unnormalized coordinate tangent vectors.

If $\lambda /a=x=m_H^2/m_0^2$ and $c/a=A+x$, then $\gamma =[72{\pi }^{2}x(A+x){]}^{-1}$ and ${\varphi }^{\ast }/T^{\ast }=(6\pi x{)}^{-1}$. Requiring both $\gamma <1$ and ${\varphi }^{\ast }>T^{\ast }$ gives $x_{-}<x<1/(6\pi )$, where $x_{-}=[-A+\sqrt{A^2+1/(18{\pi }^2)}]/2$. This is an open interval for the specified model. Failing the strength condition does not alone eliminate its first-order jump.

An electron with kinetic energy $\alpha m_e{c}^2$ annihilating with a stationary positron into two equal-angle photons has $\tan \varphi =\sqrt{2/\alpha }$ for $\alpha >0$. Equal photon energies follow from the specified symmetric laboratory geometry.

Defocus-dependent edge fringes caused by propagation and interference of the electron wave. Their apparent width and sign can change with defocus, so they do not directly measure the thickness of a material shell.

Electrical conductivity arising from motion of ions. In a dilute, uncorrelated vacancy model, a mobile oxygen vacancy has effective charge $+2e$ and moves opposite to the oxygen ions that fill it.

For orthogonal slip and fault-normal unit vectors $s,n$, the P radiation amplitude is $2M_0(\hat{r}\cdot s)(\hat{r}\cdot n)$. Its zeros are the two planes normal to $s$ and $n$, producing four alternating polarity quadrants.

Let $V=\{ϵ<r<R\}$ with $ϵ>0$ and $\mathbf{F}=\hat{\mathbf{r}}/r^{\gamma }$. Direct differentiation gives $\nabla \cdot \mathbf{F}=(2-\gamma )r^{-\gamma -1}$. For $\gamma \ne 2$, integration against $4\pi r^{2}dr$ gives the displayed expression. Independently, the two boundary spheres contribute $4\pi R^{2-\gamma }$ and $-4\pi {ϵ}^{2-\gamma }$, since the outward normal of the shell at its inner boundary points toward the origin. For $\gamma =2$, the divergence is zero in the shell and the surface contributions cancel exactly. The excluded origin must not be silently added to this regular-domain calculation.

For an isolated growing sphere, a quasi-steady exterior diffusion equation, constant surface concentration $c_s$, and a maintained far-field concentration $c_{\infty }$ give $c(r)=c_{\infty }-(c_{\infty }-c_s)a/r$. A sharp moving boundary with interior concentration $c_p$ then obeys $(c_p-c_s)\dot{a}=D(c_{\infty }-c_s)/a$. The displayed integrated law assumes all concentrations and the diffusion coefficient are constant. It becomes $a\propto \sqrt{t}$ when the seed radius is negligible. This reservoir-fed law differs from curvature-controlled Ostwald ripening in a depleting many-particle system.