The fixed-volume equilibrium crystal shape obtained by minimizing orientation-dependent surface free energy. Facet support-plane distances are proportional to their interfacial free energies under the construction assumptions.

A cubic crystal structure with lattice sites at cube corners and face centres. Its close-packed layers are (111) planes, stacked ABCABC.

Close packing of triangular layers with ABAB stacking. Its nearest-layer geometry agrees with face-centered cubic stacking, but the further-layer registry differs.

A cocrystal is a single crystalline phase containing two or more distinct molecular components in a shared ordered structure. Its components occupy defined roles in the repeating arrangement, rather than simply forming separate crystals mixed together. A stylized $1:1$ arrangement of molecules A and B illustrates the idea without assuming that every cocrystal has that ratio.

Intermolecular forces and molecular geometry influence which combined structure is favorable. A diffraction measurement tests the shared ordering, while a phase diagram distinguishes a stable combined phase from coexistence of separate A-rich and B-rich phases. This is useful because a shared lattice can change properties such as melting behavior and mass density even though the identities of the molecular components remain recognizable. The word describes a structural phase assignment and does not by itself predict which component combinations will form one.

Crystal packing describes how constituents are arranged and make contact within a crystal. It includes both the repeated geometry and, for molecular crystals, the orientations and intermolecular contacts of the molecules. Composition alone does not determine this arrangement: the same substance can have different crystal polymorphs with different packing.

A useful geometric model replaces constituents by equal hard spheres. Hexagonal close packing and face-centered cubic packing put each sphere next to twelve nearest neighbours but arrange successive close-packed layers differently. Their local contact geometry can therefore agree while their longer-range order differs. Real molecular shapes and directional intermolecular forces supply further constraints. Packing affects mass density, available slip geometry and directional response, making diffraction a way to connect structural measurements with macroscopic properties rather than merely assign a repeating pattern.

A solid containing many differently oriented crystalline crystal grains. grain boundaries impose compatibility and act as transport paths or sinks for mobile defects.

A macroscopically planar crystal boundary characterized by a crystallographic orientation. Its Miller index specifies that orientation; visible facet area depends on surface thermodynamics and growth kinetics. A low surface energy and a slow growth rate are different properties.

A one-photon polarization qubit can conditionally rotate another mode's coherent amplitude: the horizontal branch leaves $\alpha$ unchanged and the vertical branch gives $\alpha e^{i\Theta }$. The output is generally entangled. Measuring the polarization in a superposition basis can erase the branch label and prepare a coherent superposition of the two amplitudes.

With source biased and drain grounded, a constant-interaction island has diamond edge slopes $s_{-}=-C_g/C_s$ and $s_{+}=C_g/(C_{\Sigma }-C_s)$. The latter includes the gate capacitance. A gate period for charging without an orbital gap is $\Delta V_g=e/C_g$. Combining widths and slopes determines capacitances only under a stated bias and single-island model; shell gaps must not be mistaken for a universal gate period.

For an ultrarelativistic proton of energy $E$ colliding head-on with a photon of energy $ϵ$, the invariant squared collision energy is approximately $(m_p{c}^2{)}^2+4Eϵ$. Producing a final proton and neutral pion requires at least $(m_p{c}^2+m_{\pi }{c}^2{)}^2$. Thus $E_{\mathrm{th}}=m_{\pi }{c}^2(2m_p{c}^2+m_{\pi }{c}^2)/(4ϵ)$. A broad isotropic photon background turns this threshold into an energy-loss feature depending on the photon distribution and source distance, not an absolute ceiling for all arriving protons.

At first order in Earth's rotation, vertical velocity gives east acceleration $-2\Omega \cos \lambda \dot{z}$. A launch returning to its initial height lands westward, while a drop from the launch's maximum height lands eastward with one quarter the displacement magnitude. These results assume constant effective gravity, no air drag and no horizontal velocity in the Coriolis term. The northern component vanishes at this order.

The pairing Hamiltonian is projected onto empty states in a specified shell above the inert Fermi sea. A state restricted to that shell is an eigenstate of this projected model, not necessarily of a contact interaction that remains unrestricted at all momenta.

For shell width $D$, attraction $g>0$ and constant single-spin density of states $\rho$, $1=(g\rho /2)\log [(2D+B)/B]$ gives $B=2D/[\exp (2/(g\rho ))-1]$. The pair energy relative to the free threshold is $-B$. At weak coupling this scale is nonanalytic in $g$.

Gas recycling through a planetary envelope is exchange between gas near an embedded planetary embryo and the surrounding protoplanetary disk. Gas can enter and leave the embryo's gravitational neighbourhood instead of remaining in a permanently closed atmosphere. A geometrical region influenced by the embryo therefore need not contain a fixed collection of captured material.

The relevant comparison is between replacement and cooling. If new disk gas replenishes a region faster than it can lose energy, recycling can interfere with cooling-controlled planetary envelope growth. Three-dimensional hydrodynamic simulations demonstrated such connected inflow and outflow around low-mass embedded bodies.^[ref] The outcome depends on thermal structure and flow geometry; it is not a universal prohibition on gas capture by every core.

Let $X$ take integer values with probabilities $p_n$, and let $Y_{\sigma }$ be an independent centered normal distribution with positive standard deviation $\sigma$. Conditioning on $X$ gives the probability density function of their sum as $g_{\sigma }(z)={\sum }_n{p}_n{\phi }_{\sigma }(z-n)$, where ${\phi }_{\sigma }$ is the normal density. Nonnegativity and Tonelli's theorem show that this density integrates to one. For small $\sigma$, isolated atoms become narrow smooth peaks with their original probability masses.

For each fixed real $t$, dominated convergence givessince $X/\sigma \to 0$ almost surely and ${\phi }_1$ is bounded. Thus every fixed integer-valued distribution acquires a Gaussian central profile on this expanding scale, without any moment assumption on $X$. This is not a claim about uniform relative error in the tails or about finite moments. For example, if $p_n=3/({\pi }^2{n}^2)$ for $n\ne 0$ and $p_0=0$, the expectation of $|X|$ is infinite. Since $\mathbb{E}|Y_{\sigma }|<\infty$, the triangle inequality forces $\mathbb{E}|X+Y_{\sigma }|=\infty$ too. Symmetry then gives equal infinite positive and negative first moments, so there is no ordinary mean. The second moment is also infinite. A Gaussian-looking center can therefore coexist with a heavy-tailed distribution.

With $U(N)={\sum }_{i=1}^N{ϵ}_i+(Ne-Q_g{)}^2/(2C_{\Sigma })$, the addition electrochemical potential is ${\mu }_N={ϵ}_N+(N-1/2)e^2/C_{\Sigma }-e{Q}_g/C_{\Sigma }$. Gate and lead voltages enter $Q_g$. Adjacent addition potentials differ by an orbital spacing plus $e^2/C_{\Sigma }$.

For initially no stars, $M_{\star }=\Phi t+(M_0-\Phi /ϵ)(1-e^{-ϵt})$. Adding the gas solution verifies $M_g+M_{\star }=M_0+\Phi t$. Thus equilibrium of the gas reservoir coexists with continuing linear growth of stellar mass.

For ${\dot{M}}_g+ϵM_g=\Phi (t)$, multiplication by $e^{ϵt}$ and integration gives $M_g=M_0{e}^{-ϵt}+{\int }_0^t{e}^{-ϵ(t-s)}\Phi (s)ds$. Exponential memory filters the past inflow; the constant-inflow equilibrium is one specialization.

A confocal microscope combines focused illumination with a spatially restricted detection path aligned to the same specimen location. In a scanning implementation, a laser focus moves across the specimen and a pinhole in a conjugate image plane rejects much of the emitted light originating away from focus.

A photodetector records the signal at each scan position to build the image. Repeating at different depths gives confocal optical sectioning. Closing the pinhole improves rejection but discards useful photons, so signal-to-noise ratio and section thickness trade against one another. Measuring a small fluorescent bead through focus tests the actual three-dimensional response rather than assuming the nominal pinhole setting fixes it.^[ref]

Randomly distributing $n$ species with equal fractions over $N$ sites gives $\Omega =N!/[(N/n)!{]}^n$. Stirling approximation gives $S=k_{B}N\ln n$, or $R\ln n$ per mole of sites. Configurational entropy alone does not decide whether a multicomponent alloy is a stable single thermodynamic phase.