Treat the magnetic electron lens as a paraxial thin lens. A point in the specimen sends rays through different lens positions; the lens brings these to a conjugate point in the image plane, with $1/u+1/v=1/f$ and transverse magnification $-v/u$. Conversely, waves leaving different specimen points at the same angle are focused to the same point in the back focal plane. This plane therefore contains the electron diffraction pattern, with radius $r\simeq f\theta$ for small scattering angle $\theta$. The specimen is just beyond the front focal plane, giving a magnified real image farther downstream.

The sketch assumes a thin specimen, coherent approximately parallel illumination, small angles and an ideal lens without aberrations. Quantitative diffraction intensities may require dynamical scattering; the ray sketch only establishes the plane conjugacies.^[ref]

The atomic force microscope measures the cantilever deflection $d$, converted to normal force by $F=k_{c}d$ after calibration. On approach, the force is initially almost zero, then attractive; a sufficiently steep attractive force gradient causes jump-to-contact. After contact, further approach bends the cantilever and indents the sample, giving a rising repulsive branch. With tip-surface separation increasing to the right, approach proceeds from right to left. The instability occurs when the magnitude of the attractive force gradient exceeds the cantilever stiffness. The contact-branch slope depends on both cantilever and sample compliance.

TODO: biology deferred by the user.

TODO: biology deferred by the user.

For ideal spherical micelles, $v/a_0=r/3$ and $r\le {\ell }_c$, giving $P\le 1/3$. For a long cylindrical micelle, $v/a_0=r/2$, so cylinders can accommodate larger packing parameters while keeping $r\le {\ell }_c$: roughly $1/3<P\lesssim 1/2$ favours cylinders. Relatively small head area or greater tail volume reduces positive curvature.

For a planar bilayer leaflet of hydrophobic thickness $\ell$, $v/a_0\simeq \ell$, with $\ell \le {\ell }_c$. Thus $1/2\lesssim P\lesssim 1$ favours bilayer/vesicle structures, with nearly planar bilayers toward $P\simeq 1$. The two leaflets put hydrophobic tails together and hydrophilic heads toward water. For $P>1$, inverse structures become possible. These bounds are approximate packing guides; chain stretching, head interactions and aggregate free energy determine actual phase boundaries.

An amphiphile has a solvent-compatible polar head and a hydrophobic tail region. The three geometrical parameters are the tail volume $v$, effective head-group area $a_0$ at the interface, and maximum effective tail length ${\ell }_c$. Their dimensionless combination is the amphiphile packing parameter, $P=v/(a_0{\ell }_c)$.

The head area is an effective interfacial area affected by solvent, electrostatic repulsion and composition; it is not simply a fixed atomic cross-section.

TODO: biology deferred by the user.

TODO: biology deferred by the user.

TODO: biology deferred by the user.

With ballistic conductance quantization, as a gate opens the channel, conductance plateaux at $G=N_b(2e^2/h)$ appear; successive transverse modes add one quantum, approximately $77.5\mu \mathrm{S}$ per spin-degenerate subband. Their appearance is the key low-temperature electrical signature.

Require transverse confinement with separated subbands, $k_{B}T$ and measurement-bias energy well below the subband spacing, a mean free path exceeding the channel length, and smooth low-reflection coupling to reservoirs. Correct for appreciable external series resistance. Phase coherence is needed for additional interference oscillations, but perfect phase coherence across an entire macroscopic circuit is not a separate requirement for the basic mode-counting result. A field that resolves spin splitting can produce $e^2/h$ increments.^[ref]

A larger-gap AlGaAs barrier has a higher conduction-band edge than GaAs. Donors in AlGaAs supply electrons that transfer into lower-energy GaAs states, leaving a positively charged donor region. The resulting electrostatic field bends the GaAs conduction band into a triangular interface well. Motion normal to the interface is quantized, while motion parallel to it remains approximately free, producing a two-dimensional electron gas.

The confined envelope peaks just inside GaAs and decays into the AlGaAs barrier; a triangular-well treatment gives an Airy-function-like envelope with boundary matching. An undoped spacer separates the electrons from the donors, strongly reducing ionized-impurity scattering compared with uniformly doped 3D material and enabling high mobility. Band offsets, donor supply, interface roughness and electrostatics jointly determine whether a single occupied subband is obtained.

n-type doping adds donor impurities with weakly bound electrons near the conduction-band edge; ionization supplies conduction electrons and leaves positive donor ions. p-type doping introduces acceptors near the valence-band edge, producing holes and negatively charged ionized acceptors. Silicon and germanium commonly use group-V donors and group-III acceptors; dopant behaviour in a compound semiconductor depends on its lattice site.

Doping sets carrier concentration, Fermi level and resistivity. Spatially patterned donor and acceptor regions form p-n junctions, transistor source/drain regions and electrical contacts; remote doping can supply carriers to a cleaner conducting layer. Compensation and incomplete ionization mean carrier density need not equal the total impurity concentration.

For parabolic bands, the density of states near a subband edge scales as $g_{1D}\propto (E-E_n{)}^{-1/2}$, $g_{2D}\propto \Theta (E-E_n)$, and $g_{3D}\propto \sqrt{E-E_c}$. Thus a quasi-1D wire has inverse-square-root singularities at successive transverse subband thresholds, a quasi-2D well has a staircase, and a bulk 3D band rises smoothly from zero. Each expression vanishes below its threshold.

These sketches assume extended noninteracting states with simple quadratic dispersion. Finite temperature, disorder and finite lifetime broaden measured threshold features; the ideal singularity is integrable and does not represent infinitely many states in a finite energy interval.

Let the graphene lattice constant be $a=\sqrt{3}b$, and choose direct primitive vectors ${\mathbf{a}}_1=a(1/2,\sqrt{3}/2)$ and ${\mathbf{a}}_2=a(-1/2,\sqrt{3}/2)$. Reciprocal primitive vectors satisfying ${\mathbf{a}}_i\cdot {\mathbf{b}}_j=2\pi {\delta }_{ij}$ areThe first Brillouin zone is a hexagon. Its corners K and K' are inequivalent valleys; M denotes an edge midpoint. In the nearest-neighbour model, $E_{\pm }(\mathbf{k})=\pm t|f(\mathbf{k})|$, where $f(\mathbf{k})={\sum }_{j=1}^3{e}^{i\mathbf{k}\cdot {\mathbf{\delta }}_j}$ and ${\mathbf{\delta }}_j$ are the three bond vectors. The band separation is $2t|f|$: it vanishes at K/K' and is $2t$ at M, varying continuously along each edge.

This graphene zone-boundary band separation is a local conduction-to-valence separation. Ideal graphene itself remains globally gapless because the bands touch at the corners. Circumferential quantization in a nanotube selects lines through this two-dimensional band structure; whether they pass through a valley determines the ideal metallic versus semiconducting distinction.

A carbon-containing feed adsorbs and decomposes on a catalytic metal particle, releasing mobile carbon. Carbon can migrate over the surface and/or through the particle, depending on catalyst, temperature and feedstock. Supersaturation and graphitic nucleation generate a curved cap; nonhexagonal rings provide the curvature needed to close its end. Continued carbon incorporation at the cap rim lifts or extends the cap into a tube while leaving a growth interface at the catalyst.

With strong catalyst-support adhesion the particle remains at the base; with weaker adhesion it can lift with the growing tube. Several shells can nucleate and extend to form a multiwall tube. Continued growth requires access to active catalyst sites: encapsulation by an impermeable carbon coating can terminate it. The catalyst need not be liquid in every chemical vapour deposition growth mechanism, and bulk dissolution is not the only possible carbon-transport route.