In a vibrating-sample magnetometer, a magnetized specimen vibrates sinusoidally in a nominally uniform applied field. Its motion changes the magnetic flux through stationary pickup coils, producing an induced voltage by Faraday's law of induction. The component at the vibration frequency is proportional to the sample magnetic moment times vibration amplitude and frequency, for a calibrated coil geometry.

A lock-in detector referenced to the vibration extracts this small signal. Calibrate with a known moment, stabilize the vibration amplitude and sweep the applied field to measure a magnetization loop. Dividing moment by film volume gives $M$ in $\mathrm{A}{\mathrm{m}}^{-1}$; the silicon substrate and holder background must be subtracted for such a thin film.^[ref]

For a clean, adiabatically connected GaAs channel at low temperature and small bias, opening successive spin-degenerate subbands givesConductance therefore shows a staircase of plateaux, rather than the continuously geometry-dependent classical Ohmic result. Each fully transmitted mode contributes approximately $77.5\mu \mathrm{S}$. More positive, or less negative, depletion-gate voltage opens a conventional GaAs constriction; more negative voltage closes it.

Require subband spacings larger than thermal and bias broadening and a mean free path exceeding the constriction length. The general Landauer formula is $G=(e^2/h){\sum }_{\alpha }{T}_{\alpha }$ with each spin/valley channel counted separately. A material with additional valley degeneracy need not have the same step sequence as GaAs, and imperfect transmission or contact resistance blurs the ideal plateaux.

One suitable platform is a monolayer molybdenum disulfide semiconductor. An electrostatic accumulation gate supplies conduction electrons whose motion is confined to the atomic sheet; source and drain contacts act as reservoirs. A clean insulating substrate and dielectric or encapsulation reduce disorder and prevent gate leakage. Two split gates deplete the adjacent sheet and leave a narrow conducting gap, thereby confining the in-plane transverse direction while retaining transport along the gap.

Apply a small source-drain voltage, measure the current and determine linear-response conductance while tuning the split-gate voltage and carrier density. A graphene sheet is another two-dimensional carrier system, but its lack of an ordinary gap complicates purely electrostatic confinement; etched boundaries or a suitable gapped architecture may be needed. A two-dimensional material does not by itself guarantee ballistic transport or resolved subbands.^[ref]

Let $x$ run along the wire and $y$ across it. In a separable approximation,Transverse confinement gives discrete levels $E_n^{\perp }$, but the long unconfined direction has a nearly continuous longitudinal wave vector and a plane-wave factor. The result is a family of one-dimensional subbands, not a completely discrete spectrum in both in-plane directions. For a hard-wall width $W$, $E_n^{\perp }={\hslash }^2{\pi }^2{n}^2/(2m^{\ast }{W}^2)$; an electrostatic confinement potential gives different thresholds. The suppressed out-of-plane degree of freedom is already frozen into the 2DEG subband.

Use complementary composition, spatial and population measurements:

The strength of TEM is correlating structure, size and local chemistry; its limited sampling is why a representative bulk chemical assay remains valuable.

A bulk fcc atom has twelve nearest neighbours, whereas surface atoms are undercoordinated: ideal (111) terrace atoms have nine and (100) terrace atoms eight. Steps, kinks, edges and corners can have still lower coordination and different bond-angle environments. These sites change electronic structure, adsorption energies and activation barriers, making geometrically different sites catalytically inequivalent.

In a Pt-Pd particle, a site's neighbours may be Pt, Pd or a mixture; surface segregation means surface composition need not equal overall composition. Alloying also changes strain and electronic interactions. Surface relaxation, reconstruction, adsorbed ligands, oxide or contamination can further alter accessibility and chemical state. A high nominal Pt fraction therefore does not establish either accessible Pt sites or a particular catalytic activity.

In cluster-beam deposition, evaporated or sputtered cobalt atoms cool in an aggregation gas, nucleate clusters and are carried to a substrate. Source temperature, gas pressure, aggregation length and residence time tune the mean size and distribution. A mass selector can narrow the delivered size range, at the cost of transmitted flux; an unfiltered beam does not automatically produce monodisperse particles.

Shape control is less direct. Growth kinetics and cluster temperature can favour particular structures, while multiple morphologies can coexist at one size. Landing energy, substrate interactions and post-deposition annealing can change the delivered shape or cause coalescence. Soft landing helps preserve gas-phase clusters. In solution, ligands, selective facet binding, seed-mediated growth and precursor feeding can give stronger chemical control over shape and size, but introduce ligand/solvent residues and require purification. Neither method guarantees a unique shape merely by selecting particle size.

The Winterbottom construction extends the Wulff construction to a crystal supported on a substrate. It includes the particle-substrate interfacial energy and the substrate surface energy displaced by contact, so the equilibrium Wulff body is truncated by the support plane. The relevant contact contribution is $({\gamma }_{ps}-{\gamma }_s)A_{\mathrm{contact}}$; it determines wetting and contact geometry. The basic unsupported construction accounts only for orientation-dependent free surface energies.^[ref]

A decahedral nanoparticle is a multiply twinned nanoparticle: five fcc-like tetrahedral domains share a common fivefold axis and meet on coherent twin planes. The angle of an ideal fcc tetrahedral wedge is $\arccos (1/3)=70.53^{\circ }$, so five wedges leave about $7.36^{\circ }$ of angular deficit. Elastic distortion and defects accommodate closure. The external shape can be a pentagonal bipyramid or a truncated variant; it is not one unstrained, periodically repeated fcc crystal.

An icosahedral nanoparticle comprises twenty tetrahedral domains arranged around a common centre, with twenty approximately triangular outer facets, twelve vertices and thirty edges. Five domains meet around each radial fivefold direction. Coherent twinning exposes many low-energy close-packed facets, but fivefold symmetry cannot tile ordinary three-dimensional space periodically without strain. The competition between favourable surface/twin energies and volume-dependent elastic strain explains why such morphologies are especially plausible for small particles.

These labels describe the observed domain geometry; cobalt can adopt different crystal phases, so identifying fcc-like twinning also requires diffraction rather than assuming every cobalt particle is bulk fcc.

For a positive-tone opening, place small clear serifs beyond convex outer corners and extend the ends of the L arms to compensate for line-end shortening. Adjust the concave inner corner with a local inward boundary correction or opaque notch where excess exposure would otherwise round/fill it. The diagram exaggerates these additions and the inner correction for visibility; their dimensions are subresolution corrections, not intended printed protrusions. In an opposite mask/resist polarity the clear/opaque roles reverse. An L with corrected ends, outer-corner serifs and an independently corrected inner corner is the appropriate mask concept. The particular bias must be fitted to the aerial image plus resist threshold over the required focus/exposure window. A printed test L, measured at the inner and outer corners, is more informative than assuming any decorative serif produces an exact L.

Treat the letter-shaped regions as openings to be etched, using positive-tone resist; the clear mask areas therefore correspond to those openings. Label the two L features $L_1,L_2$ from left to right. An alternating assignment is mask A: H, $L_1$, O; mask B: E, $L_2$. Each full letter remains on one mask, avoiding unnecessary stitch boundaries across its stems. Nearest-neighbor interletter gaps now occur between different exposures, rather than having to be resolved within one exposure. Verify that each individual letter's internal spaces are still printable; without dimensions, this partition is a process proposal rather than proof of a particular numerical minimum linewidth.

A robust sequence is litho-etch-litho-etch double patterning into a durable hard mask. Deposit an oxide hard mask on silicon; expose and develop mask A and transfer those openings through the hard mask. Strip resist, apply fresh resist, align and expose mask B, then transfer its openings into the same hard mask, the second resist covering mask-A openings during that second oxide transfer. Remove resist and perform one final deep silicon etch through the union of the openings. An inductively coupled deep reactive-ion etching process with alternating fluorine etching and sidewall passivation gives much more vertical, high-aspect-ratio features than an isotropic etch. Mask thickness and etch selectivity must survive the required depth. Control sidewall scalloping, aspect-ratio-dependent etch rate, charging, microloading and the combined overlay/etch-bias budget.

A spacer-based process can produce very fine regular pitches, but arbitrary letter junctions and terminations require additional cut/block patterning; it is not automatically a two-mask solution for this target. Use two independently transferred sparse masks, then a selective directional deep etch. The schematic gives the assignment and a generic corner-correction example without reproducing the supplied pattern.

Finite numerical aperture and diffraction remove high spatial frequencies from a mask image. Neighboring features modify the aerial intensity, and the photoresist threshold turns this blurred intensity into biased edges, shortened lines and rounded corners. optical proximity correction deliberately predistorts the mask: line-end extensions and small serifs can compensate for local loss of exposure, while other edges are moved back where exposure spreads too far. The correction is calibrated for illumination, resist and focus; it does not restore arbitrarily high lost spatial frequencies.

double patterning separates a dense target pattern into two sparser patterns, each printable within the single-exposure process window. Two exposures with an intervening pattern-fixing or transfer step combine their features on the wafer. For regular lines, alternating line assignment can halve the final pitch while each exposure retains a larger pitch. optical proximity correction corrects local image bias; double patterning divides the density burden between exposures. Overlay error and the minimum printable isolated feature still constrain the result. Simply exposing the same unmodified resist twice without an appropriate process separation does not give the same independent thresholding.

Indexed electron diffraction ring positions from a polycrystalline deposit identify candidate phases and give lattice parameters after camera-length calibration. Spot patterns from a small selected area establish zone axis directions, epitaxial orientation relationships or crystal twinning. Nanobeam mapping can reveal grain orientation changes, while systematic spot shifts can measure relative strain if lens distortions and scan drift are controlled. Spot or ring broadening can reflect finite coherent size, but also strain and instrumental broadening, so it is not an unqualified particle-diameter measurement. Use a calibration standard and an empty support region, vary dose to check electron beam damage, and combine spacing information with elemental analysis when chemically distinct phases have similar lattice parameters.

Choose cyclic reciprocal units, so the incident and elastically scattered wavevectors have magnitude $1/\lambda$. Place the reciprocal lattice origin $O$ at the end of the incident vector from the sphere centre $C$. A reciprocal vector $\mathbf{g}$ is excited when its endpoint $G$ lies on the Ewald sphere, becauseA specimen tilt rotates the reciprocal lattice relative to this construction. The very short electron de Broglie wavelength makes the sphere nearly flat across several ordinary reciprocal-lattice spacings. In a thin foil, finite thickness $t$ broadens reciprocal points into reciprocal lattice rods approximately along the foil normal, with characteristic longitudinal extent $1/t$. A rod may intersect the sphere even when its ideal point is slightly off it. Consequently several reflections can appear simultaneously near a zone axis, and thinning changes the excitation tolerance. A small particle has finite-size broadening in other directions too. This geometrical excitation condition does not guarantee a particular spot intensity: crystal basis structure factors, specimen thickness and dynamical diffraction still matter.

In transmission electron microscopy, the electron objective lens forms an intermediate real-space image and intermediate/projector lenses enlarge it onto a screen or camera. The total optical magnification is the product of the relevant stage magnifications. The user changes imaging-lens excitations to select magnification, while separately controlling illumination and focus. Display resizing introduces an additional enlargement, so the recorded pixel calibration matters.

In scanning transmission electron microscopy, a focused probe is rastered across the specimen and a detector signal is assigned to each probe position. The displayed magnification is display width divided by scanned specimen width. The user increases magnification by reducing raster-deflection amplitude, independently of choosing a smaller incident probe with the probe-forming lenses. TEM changes image-forming lens magnification; STEM changes the raster field of view. A smaller field or more pixels does not create finer physical resolution than the probe, specimen scattering and detection permit. A calibrated real-space scale is more transferable than the apparent size on a monitor.^[ref]