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The stoichiometry fixes the site probabilities to $1/4$ gold and $3/4$ copper. Every site in the average crystal structure therefore has the same atomic scattering factorThe average structure factor isAt $(110)$ the bracket vanishes, while at $(111)$ it is four. HenceThe $(111)$ powder diffraction multiplicity is unchanged, so its coherent Bragg diffraction wave intensity equals the ordered value under the stated equal-lattice constant approximation.

The zero refers to the coherent Bragg diffraction peak of the average crystal structure. A random alloy can still give occupational diffuse scattering at the same scattering vector: the average of the squared amplitude includes fluctuations as well as the square of the average amplitude. Such background is not a surviving $(110)$ Bragg diffraction peak.

Specify the normalization of the alloy sublattice order parameter rather than assuming that its endpoint values uniquely define it. Let $p_c$ be the probability of gold occupation at a corner site and defineThe complementary copper probability is $3(1-\rho )/4$, so the corner-site atomic scattering factor isThe stoichiometry provides one gold chemical atom per four-site unit cell. If all three face sublattices are equivalent, their gold occupation probability $p_f$ obeys $p_c+3p_f=1$. ThereforeAt $\rho =1$ the corner and face sites carry pure gold and pure copper; at $\rho =0$ they have identical statistical compositions. These formulas describe the branch with $0\le \rho \le 1$ and equivalent face-site averages.

Sample A is the Cu–Au alloy. At the indicated copper-rich composition, the supplied Cu–Au diagram permits one copper-rich solid-solution phase. The Cu–Ag diagram instead gives two equilibrium solid phases at room temperature. A single consistent set of FCC peaks therefore identifies A under the stated equilibrium assumptions; similar pure-metal lattice parameters alone would not distinguish gold from silver.

Use Vegard's law as an approximate atomic-fraction interpolation:With the higher-angle estimate from part(a),This is about $5.1$ atomic percent, not weight percent. Using rounded molar masses $M_{\mathrm{Au}}\simeq 197$ and $M_{\mathrm{Cu}}\simeq 63.55\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$,Using the first peak alone instead gives roughly $12.6wt\%$. The justified conclusion is therefore about $13$–$14wt\%$ gold, with uncertainty from angle rounding, calibration and the approximate linear lattice-composition relation.

SEM with energy-dispersive X-ray spectroscopy can identify gold versus silver through their characteristic X-ray energies. Acquire spectra and maps from representative regions: two compositionally distinct regions would support the two-phase interpretation of B. With standards or validated matrix corrections it can estimate local compositions and, with representative sampling, an overall composition. Surface topography and the interaction volume limit spatial interpretation.

X-ray fluorescence provides a complementary more bulk-averaged elemental measurement. Gold and silver produce different characteristic lines; calibrated intensities can yield their mass fractions, with corrections for absorption, excitation and specimen geometry. It does not by itself determine whether an element occupies one or two crystallographic phases.

Both methods supply elemental information missing from a lattice-parameter-only identification. A purely morphological micrograph could reveal phase separation but would not alone give a reliable numerical chemical composition.

In a suitably dilute, nonoxidizing chemical acid, hydrogen sulfide precipitates the sufficiently insoluble sulfides, including those of copper, lead, Mercury, bismuth, cadmium, arsenic, antimony and tin. “Sulphuretted hydrogen” is the historical name for hydrogen sulfide, ${\mathrm{H}}_2\mathrm{S}$. Oxidation state and acidity must be specified: the statement is not independent of every possible chemical solution condition.

Arsenic is now classed as a metalloid; the historical analytical grouping was not a rigorous modern division into metals and nonmetals. Dissolved silver also gives black ${\mathrm{Ag}}_2\mathrm{S}$ in acidic conditions. It is often absent from the conventional second analytical group because a preceding hydrochloric acid treatment has already removed it as silver chloride. Suitable gold and platinum solutions can likewise yield dark sulfide precipitates, although their speciation and reduction chemistry make a simple universal colour rule inappropriate. Thus an analytical group is a separation convention, not a list of every ion that can possibly react.

For a divalent metal ion, a useful net balance isThe selectivity follows from two coupled chemical equilibrium conditions. Hydrogen sulfide is a weak acid; increasing $[{\mathrm{H}}^{+}]$ suppresses its ionization. Approximately,Even this small sulfide activity can exceed the solubility product of very insoluble sulfides. More soluble sulfides, such as those of zinc and manganese, generally require less acidic or alkaline conditions. Consequently an excessively strong chemical acid, complex formation or an oxidizing medium can change the result. Colours alone are preliminary indications, not final identification.^[ref]

The traditional preparation treats iron(II) sulfide with a dilute nonoxidizing chemical acid:A gas-generating flask with controlled acid entry and a delivery outlet conducts the product to the test solution. The apparatus must permit gas flow and contain the effluent rather than accumulating pressure in a sealed vessel. Sulfuric acid in dilute form can replace hydrochloric acid; concentrated oxidizing acids are unsuitable for this simple preparation. Hydrogen sulfide is acutely toxic, and loss of odour perception makes smell an unreliable warning: the historical apparatus describes a laboratory principle, not a household test.^[ref]

nickel is less noble than copper and can dissolve anodically, but its reduction requires a much more negative cathodic electrode potential. It therefore normally accumulates in solution rather than codepositing with copper. lead can also oxidize, but sulfate can precipitate the resulting lead ions as poorly soluble lead sulfate, contributing to anode slime or passivation. gold normally remains in the solid residue. silver is normally recovered predominantly in anode slime, but its path can include dissolution and reprecipitation or cementation: a permanent refusal of every silver atom to dissolve is not the mechanism. Experiments with controlled Cu–Ag–Pb anodes, a fixed sulfate electrolyte and SEM/EDS plus diffraction analysis explicitly find metallic silver and sulfate-containing slime phases.^[ref]

Selective electrode potential, solution management and physical exclusion of slime permit purer copper deposition. However, the unusually large printed anodic slope prevents an unconditional silver-retention claim here. The preceding calculation places the anode near $0.34+0.6788=1.0188$ V versus the standard hydrogen electrode, above the tabulated ${\mathrm{Ag}}^{+}$/Ag value. silver oxidation is then thermodynamically possible; any silver ions reaching the cathode can be reduced preferentially to copper. Actual activities, precipitation, cementation and transport determine contamination. The requested 99.999 wt% purity cannot be established from the listed standard potentials alone. Monitor anode potential against a reference, assay the electrolyte and deposit, and prevent slime entrainment; do not silently replace the printed 0.295 V/decade with a smaller number to remove this issue.

First hold the production rate fixed, so total electric current remains $200\mathrm{A}$. If only the impure anode receives a factor $10^5$ increase in accessible area, its local current density becomes $0.002\mathrm{A}{\mathrm{m}}^{-2}$, far below $j_0=1\mathrm{A}{\mathrm{m}}^{-2}$. Extrapolating the large-current Tafel equation would give a negative anodic overpotential, which is physically the wrong regime. Near equilibrium the Butler-Volmer equation instead gives, in the same conditional two-electron model,The anode activation loss nearly disappears; the 1.20 V ohmic loss does not disappear. The cathode still operates close to its original transport limit if it is unchanged. The conditional kinetic model gives $V_{\mathrm{app}}\simeq 1.3286$ V for anode-only enlargement. Its anodic potential returns close to 0.34 V, making retention of noble silver and gold more favorable, while less noble nickel dissolution and lead sulfate formation remain relevant.

If both active areas increase by $10^5$ and their independent mass-transfer capacity scales by the same factor, both local kinetic losses become small and the conditional prediction is about $1.20005$ V. Porous internal area does not generally multiply an external diffusion-limited flux by $10^5$: pore access, ion depletion, tortuosity, electrical connectivity and slime blockage must be checked. If the specified 200 A/m² is instead held per true enlarged area, total current rises to $2\times 10^7$ A and the unchanged electrolyte resistance would give $1.2\times 10^5$ V, so this is an entirely different operating condition.

For the same refining rate and the same effective kinetic model, the anode-only estimate is therefore $1.3286\mathrm{V}$. The robust conclusion at fixed total current is reduced local activation polarization on the genuinely accessible enlarged surface, with an unchanged ohmic floor unless cell geometry or conductivity also changes. Increased geometric area alone does not prove a proportionate decrease in energy cost or improved purity. The distinct area and current conventions above specify how to interpret the prediction; they must not be mixed in one calculation.

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.

The resolved periodic contrast comes principally from electron phase contrast: transmitted and diffracted electron waves interfere after passing through the image-forming optics. The resulting intensity depends on specimen thickness, orientation, defocus and the contrast transfer function; bright dots cannot automatically be labelled gold-atom positions without image simulation. The surrounding nonperiodic background is consistent with an amorphous support or contamination, but its chemistry is not determined by contrast alone.

The particle contains differently oriented fringe regions, and the Fourier transform has several opposing spot pairs with an approximately pentagonal/tenfold composite appearance. Together with the faceted outline, this is consistent with a multiply twinned gold particle, plausibly a decahedral arrangement, rather than one perfect cubic lattice with fivefold rotational symmetry. Opposite Fourier spots occur because the recorded image intensity is real; counting ten spots is not itself proof of a tenfold crystal. Overlapping domains, contrast-transfer effects and a slightly off-axis view can complicate the pattern. A tilt series or spatially windowed transforms across individual domains would test the twinning interpretation. The stability of such particles involves a competition among surface, twin-boundary and elastic strain energies.^[ref]

Locate an isolated nanowire first, then align source, drain, dielectric and top-gate patterns to it. A suitable research-device sequence is the following; the semiconductor is unspecified, so contact-barrier and surface-cleaning details cannot be assumed universally ohmic.

Prepare a clean electrically insulating substrate, with alignment marks. Disperse a sparse population of nanowires from the suspension, remove the carrier solvent compatibly, and locate a suitable isolated wire by microscopy. Record its position relative to the marks.

Use aligned electron beam lithography and a suitable lift-off process to open contact regions near the two ends while leaving a central channel. Clean the exposed contact regions without destroying the wire, evaporate gold and lift off. The contacts overlap the ends but do not bridge the central channel; test their electrical behavior before assuming an ohmic response.

In a second aligned pattern, define a central dielectric patch that extends beyond the proposed gate edges. Evaporate the specified silicon monoxide and lift off using a profile that permits film separation. Verify coverage over the wire and absence of pinholes: directional deposition onto curved nanoscale geometry is not automatically conformal.

Define and evaporate a gold top gate over that insulated central patch, with its lead kept electrically separated from the source, drain and wire. Lift off and test gate leakage before extracting transistor characteristics. The dielectric must remain continuous under the gate and overlap its edges; the illustrated central gate is not a third direct metal contact to the semiconductor. Gold and silicon monoxide are used as stipulated, without silently replacing SiO by a different oxide.

Use a self-assembled monolayer of an alkanethiol with a polar terminal group, such as $\mathrm{HS}(\mathrm{C}{\mathrm{H}}_2{)}_{\mathrm{n}}\mathrm{OH}$. Sulfur anchors to gold while the exposed hydroxyl groups interact favorably with water. A methyl-terminated monolayer would instead present a low-polarity outer surface and usually retain hydrophobic character.

Prepare a clean gold surface, allow adsorption from a compatible clean solution, and rinse away unbound material. Dense coverage and molecular ordering help make the terminal chemistry reproducible. Mixed chain lengths or incomplete coverage can change the wetting response, so the presence of an OH group in the bottle alone is not sufficient. An original experiment demonstrated hydrophilic hydroxyl-terminated films and showed that mixing chain lengths changed surface order and wettability.^[ref]

This specifies the resulting interface rather than treating every freshly prepared gold surface as intrinsically identical: contamination, preparation and adsorbates influence the starting contact angle.

A useful net bookkeeping reaction for thiol adsorption iswith removal or reaction of the hydrogen depending on environment. Equivalently, a simplified overall balance can combine two hydrogen atoms into ${\mathrm{H}}_2$. These balances identify sulfur anchoring; they are not a claim that one elementary adsorption step universally releases hydrogen gas.

Molecules arrive at the interface, adsorb through sulfur, and reorganize laterally as coverage grows. Interactions between the chains favor packing and order, with the hydroxyl end groups facing outward. Gold surface structure, defects and adsorption conditions affect the final film.

Use a shaped nickel particle or nanowire as both sacrificial template and reducing agent. gold grows where nickel is consumed, allowing gold-coated particles, porous shells and, with suitable removal of the remaining nickel, hollow gold nanostructures. The original template controls the overall shape; the balance of dissolution, deposition and diffusion controls wall thickness and porosity.

A hollow product is not guaranteed simply by writing the reaction: deposition may seal a solid nickel core, whereas defects can maintain access to the interior and permit further replacement. Controlled composition and transport can give a progression from core–shell to porous or hollow structures. This is useful for high-surface-area electrodes, catalytic surfaces and plasmonic structures, while avoiding an electrical lead to every individual nanoparticle.

Clean and degrease the nickel surface, remove its inhibiting oxide by an appropriate brief surface treatment, rinse, and immerse it in an aqueous bath containing reducible gold(III) species. Control concentration, temperature and immersion time, then rinse after the desired deposition. A solution of a gold complex is practical, but its actual reduction potential must be evaluated for that complex and bath: it need not equal the supplied free-ion standard potential.

At anodic sites nickel dissolves as $\mathrm{N}{\mathrm{i}}^{2+}$, releasing electrons into the conducting substrate. Those electrons reduce gold ions at cathodic sites, nucleating and growing metallic gold. The nickel itself is the reducing agent, so no external electrical supply or separate reducing reagent is required. The coupled electron balance is three nickel atoms dissolved for two gold atoms deposited for the stated oxidation states.

The process can slow or stop when a sufficiently continuous gold coating prevents the electrolyte from reaching nickel. This is not an absolute thickness rule: pores, cracks or exposed edges preserve nickel dissolution and allow continued deposition, sometimes with undercutting. nickel passivation, depletion of gold ions near the surface, diffusion through a porous layer, bath speciation and competing reactions also affect the rate. Dense gold coverage and porous gold coverage therefore have very different limiting behavior.

The species that oxidizes nickel is gold(III) ion, rather than metallic gold. The half-reactions areMultiplying the first by three and the second by two givesStandard electrode potentials are not multiplied by the balancing factors. Thus $E_{\mathrm{cell}}^{\circ }=1.50-(-0.25)=1.75\mathrm{V}$ and six moles of electrons are transferred per mole of this balanced reaction:The negative sign predicts a thermodynamically favorable galvanic replacement under standard conditions.

Use the supplied polynomial as an effective-atomic-number approximation for the alloys. Interpreting each quoted phosphorus number as an atomic fraction $x_P=N_P/(N_P+N_{\mathrm{Ni}})$,The gold reference yield is $\eta (79)=0.486996$. Substitution gives:

The ordering is gold brightest, then copper, NiP${}_z$, NiP${}_x$ and NiP${}_y$. The phosphorus-rich layer has the lowest effective atomic number.

There is a wording ambiguity in “atomic fraction of P relative to Ni.” If those numbers instead mean $r=N_P/N_{\mathrm{Ni}}$, first convert $x_P=r/(1+r)$, giving $\bar{Z}=(28+15r)/(1+r)$. The three alloy contrasts then become $0.4243$, $0.4361$, $0.4129$, respectively. The qualitative ordering is unchanged. Both calculations explicitly use $\eta (\bar{Z})$; this simple alloy approximation is not a full electron-transport simulation.

In high-angle annular dark-field imaging scanning transmission electron microscopy, electrons scattered to large angles give predominantly incoherent mass–thickness and atomic-number contrast. For comparable thickness, gold is much brighter than nickel–phosphorus because its atomic number is much larger. carbon in the protective cap scatters weakly, whereas its tungsten component scatters strongly. The cap is not a homogeneous low-atomic-number reference.

The gold layer follows a curved, locally distorted interface. The dark central opening in the overview is consistent with a crack, void or locally missing/thinned foil: little material along the beam path gives little high-angle scattering. The granular and columnar variations beneath the gold indicate nanoscale heterogeneity in the affected NiP${}_y$ region. Differences in thickness, density, porosity and composition can all contribute; diffraction/channeling effects are reduced in high-angle annular dark-field imaging but not eliminated in every geometry.

A black region is therefore evidence of low projected scattering mass, not by itself the identity of a chemical phase. In particular, a single high-angle annular dark-field imaging image does not distinguish a pre-existing void from material lost during specimen preparation.

The thin scanning transmission electron microscopy foil restricts scattering to a much smaller projected specimen volume than bulk SEM at $30\mathrm{kV}$. It reveals the nanoscale shape and continuity of the gold/NiP interface, the narrow low-density opening and the fine heterogeneous region underneath the gold, which the larger SEM interaction volume and surface-sensitive collection obscure.

The enlarged view resolves structure on the tens-of-nanometres scale: local interface roughness, small bright and dark domains and the relationship of the gold layer to the damaged or porous-looking NiP${}_y$ region. This helps distinguish an interface-associated defect from merely a change between nominal alloy layers. It still does not identify a crystalline phase or quantify composition without diffraction or spectroscopic analysis. Since the SEM surface and focused ion beam milling lamella need not cut precisely the same volume, correlations should allow for section position.

The contact dimensions are readily accessible to UV photolithography followed by directional evaporation and lift-off. A possible sequence is:

A real device must also have appropriate ohmic contact metallurgy. gold alone on arbitrary GaAs is not automatically an ohmic contact: surface preparation, heavy contact-layer doping and, where required, a suitable alloyed contact stack are needed. The schematic gold label specifies the visible electrode, not all such process details. Order the mesa etch and contact steps so that the etchant and any anneal do not damage an already formed contact.

Prepare size-characterized gold-particle ensembles, heat them on a calibrated in-situ electron diffraction stage, and follow the loss of sharp fcc diffraction features and appearance of liquid-like diffuse scattering. Measure the particle size distribution rather than treating a polydisperse ensemble as one diameter. Temperature-dependent weakening of a diffraction peak alone is insufficient: thermal vibrations, particle motion, coalescence, evaporation and beam damage can also reduce intensity. Low-dose repeats, heating/cooling runs, size checks before and after heating, a bulk-like reference and support-dependent controls help distinguish melting from those effects. Electron diffraction measurements on gold particles down to about 2 nm are a classic demonstration.^[ref]

The physical origin is the large interface contribution to free energy at small radius. A curved solid–liquid interface raises the solid's thermodynamic equilibrium chemical potential relative to the flat-interface case, so coexistence occurs below the bulk melting temperature. Reduced coordination and surface disorder provide a microscopic interpretation, but the quantitative continuum relation is a free energy balance, not simply a claim that every surface atom has a universal lower melting point.