Let $(a_1,a_2,a_3,a_4)=(0.1,0.4,0.3,0.86)$ and $S=0.9996$. For the time-independent well,It remains zero outside the well. Each expansion coefficient acquires a phase of unit magnitude, so every energy probability, including the preceding $26.0\%$, is time-independent. The position probability density generally changes because cross terms between different energies oscillate.

Only $n=1,2,3$ have $E_n<10c$. The printed squared coefficients sum to $S=0.1^2+0.4^2+0.3^2+0.86^2=0.9996$, rather than exactly one. Normalizing those numbers givesTreating the coefficients as rounded values of an already normalized state gives the intended approximately $0.26$ answer. Probabilities are sums of squared amplitudes because the energy eigenstates are orthonormal.

Mesa definition comes first so that the two front contacts can be aligned to an already isolated active area and can avoid damaged mesa edges. Keeping the cap until after metallization provides the intended contact surface and protects the window during the contact process. lift-off process must precede selective cap removal if the retained metal is to act as that etch mask. The cap etch must stop above the emitter rather than destroy the junction.

Rear metallization after the wet-processing stages avoids unnecessarily exposing the rear contact to their etchants. Contact activation precedes the optical coating where possible, because the anneal can alter dielectric properties and optical thickness. The antireflection coating goes last and leaves the pads uncoated for electrical access. The key ordering constraints are clean semiconductor before contacts, electrical isolation without severing the intended current path, selective aperture opening, and a thermal budget compatible with every retained layer. If a different metal or etchant requires a changed order, it must preserve these same functions.

Use optical photolithography for the two masks and a resist profile compatible with the selected process. A controlled wet mesa etch or a calibrated reactive-ion etching process must pass completely through the p–n interface, with limited sidewall damage. Use profilometry or cross-section controls to verify depth and a test device to check leakage. For the front lift-off process, directional evaporation and an undercut resist profile discourage a continuous metal fence bridging the mask sidewall.

The cap-removal step requires a separately qualified selective etchant: its selectivity between gallium arsenide and the particular aluminium gallium arsenide alloy depends on alloy composition, chemistry and processing conditions. Establish etch rate and endpoint on a witness stack; a universal etch time cannot be inferred from a diagram without thicknesses. The front metal must resist that etchant. Back metallization and activation must establish an ohmic contact rather than an unintended rectifying junction. Verify specific contact resistance on test structures and avoid an anneal that degrades the p–n profile. The final dielectric can be deposited by a low-damage evaporation or other compatible thin-film method.

One workable route uses two resist coatings and a metal-defined optical aperture. The following are eleven fabrication operations; exposure/development is grouped as one lithographic operation, and rear metallization includes its contact activation if required. Detailed cleaning, rinsing and inspection occur between compatible operations.

1. Spin-coat a first photoresist on the front surface to define a mesa mask.
2. Expose and develop that mask, retaining resist over the complete active-cell mesa, including its future front contacts.
3. Etch the exposed semiconductor through the p-type layers and slightly into the n-type base. This removes the unwanted junction around the cell and produces the lateral mesa step.
4. Strip the first photoresist, leaving a clean mesa with its cap intact.
5. Spin-coat a second photoresist suitable for the lift-off process.
6. Expose and develop openings at the intended front grid or contact pads, positioned safely inside the mesa edge.
7. Evaporate a p-contact metal stack onto the exposed gallium arsenide and resist. Au/Zn-based contact metallization is a conventional candidate, subject to doping and contact-resistance qualification.
8. Dissolve the second photoresist to lift off unwanted metal, leaving the front contacts on the cap.
9. Selectively remove the cap from the light-receiving aperture, using the front metal as a hard mask and the aluminium gallium arsenide window as the etch stop.
10. Deposit the rear n-contact metallization, for example a qualified AuGe/Ni/Au system, with an appropriate contact-activation anneal if required and compatible with the front contact.
11. Deposit an antireflection dielectric over the optical aperture through a mechanical shadow mask that keeps the contact pads accessible. Its refractive index and thickness are chosen for the target optical band.

The essential material changes are junction isolation at the mesa edge, metal retained on the front cap, cap removed only from the optical aperture, and a conducting rear contact. The metal systems and anneal are a process design, not compositions specified by the drawing. A workable design must also qualify the passivation of etched sidewalls and the front-contact surface preparation.

For $D=0.001$, the same duty cycle model gives $\bar{I}=25\mu \mathrm{A}+0.999(6.1\mu \mathrm{A})=31.0939\mu \mathrm{A}$. Thus $\bar{P}=62.1878\mu \mathrm{W}$. The tenfold reduction in active fraction reduces average power by only about $8.23$ because the sleep current remains. The ideal limit as $D\to 0$ is $V{I}_s=12.2\mu \mathrm{W}$, before leakage elsewhere or wake-up energy. Active and sleep contributions become comparable at a duty cycle of roughly $I_s/(I_a+I_s)=0.0244\%$; below that scale lowering sleep current is particularly valuable.

The duty cycle $D=t_{\mathrm{active}}/(t_{\mathrm{active}}+t_{\mathrm{sleep}})$ is the fraction of time spent active. At a constant supply voltage and with switching energy neglected,For $D=0.01$, the average current is $0.01(25\mathrm{mA})+0.99(6.1\mu \mathrm{A})=256.039\mu \mathrm{A}$. Thus $\bar{P}=512.078\mu \mathrm{W}\simeq 0.512\mathrm{mW}$. Multiplying only the active power by the duty cycle would omit the sleep contribution.

A $0.10\mathrm{M}$ formula-unit concentration of fully dissociated $\mathrm{AlC}{\mathrm{l}}_3$ gives $0.10\mathrm{M}$ trivalent cations and $0.30\mathrm{M}$ chloride:At unchanged temperature and permittivity, the Debye–Hückel screening length scales as $I^{-1/2}$, giving ${\kappa }^{-1}=0.96/\sqrt{6}\simeq 0.39\mathrm{nm}$. It falls by a factor $\sqrt{6}$, not by three. This is a formal screening estimate: at this ionic strength, trivalent-ion activity, finite ion size, hydrolysis and specific adsorption make the ideal dilute model quantitatively limited.

For an ideal dilute electrolyte with molar ionic strength $I=\frac{1}{2}{\sum }_i{c}_i{z}_i^2$, measured in $\mathrm{mol}{\mathrm{L}}^{-1}$, the Debye–Hückel screening length isTaking water's relative permittivity as approximately $78.5$ at $25^{\circ }\mathrm{C}$, $I=0.10\mathrm{M}$ for NaCl. Therefore ${\kappa }^{-1}\simeq 0.96\mathrm{nm}$. This is an exponential screening length, not a hard outer edge of the diffuse layer. The factor of $1000$ converts molarity into $\mathrm{mol}{\mathrm{m}}^{-3}$.

The change from a positive surface potential to a negative potential outside the compact layer is consistent with specific phosphate adsorption that overcompensates the positive oxide charge. The negative compact-layer charge then has a diffuse excess of positive counterions. This charge reversal is an interfacial-chemistry interpretation, not something a single monotone double-layer profile with an unchanged positive boundary charge could generate by screening alone. Surface potential and zeta potential at a slip plane are distinct observables.

The Stern model separates the specifically adsorbed compact layer from a diffuse layer. In a spherical diffuse layer, a Poisson-Boltzmann equation description isAt sufficiently small $|z_{i}e\psi |/k_{B}T$, the Debye–Hückel approximation gives ${\nabla }^2\psi ={\kappa }^2\psi$. If $a_s$ is the outer compact-layer radius, its decaying solution isFor a locally flat surface, the geometric factor tends to unity. Thus the far-field negative potential approaches zero from below over the Debye–Hückel screening length. Neither $+120$ nor $-70\mathrm{mV}$ is safely in the low-potential linear regime at room temperature, so the simple exponential is a far-field approximation, not a globally valid fit. The phosphate layer requires adsorption chemistry or an equivalent specified boundary model.

A Pickering emulsion is stabilized by particles adsorbed at the liquid–liquid interface. A partially wetting sphere removes a patch of that interface. The smaller detachment barrier into either bulk phase is approximatelywhere $\theta$ is the three-phase contact angle. At approximately neutral wetting, $\theta =90^{\circ }$, the stated particle radius givesat $25^{\circ }\mathrm{C}$. Such a large attachment energy makes thermal detachment unlikely. A sufficiently covered interface resists drop coalescence, and a water-wet particle preference supports an oil-in-water dispersion. However, radius alone does not guarantee stabilization: completely water-wet or oil-wet particles have a vanishing attachment barrier in this model, and sufficient particle coverage is necessary. Sedimentation and droplet ripening are separate failure modes. The classical particle-stabilized-emulsion experiments provide the historical basis of the terminology.^[ref]

An amphiphile adsorbs at the oil–water interface, reducing the interfacial tension. At fixed drop radius and oil volume, $E_s(c)=A{\gamma }_{ow}(c)$ therefore falls with increasing concentration and tends toward a plateau once the interface is nearly saturated and further amphiphile mostly enters micelles. For an approximately ideal dilute nonionic surfactant, the Gibbs adsorption equation $d\gamma =-RT\Gamma d\ln a$ connects the fall in interfacial tension to positive surface excess $\Gamma$. Ionic systems require the appropriate counterion activities.

The critical micelle concentration gives the familiar break in a tension-versus-log-concentration curve, rather than a fall to zero interfacial energy. If mixing creates a different droplet size distribution as amphiphile is added, area also changes; a tension reduction alone does not determine the total interfacial energy. The plotted prediction holds area fixed.

For identical spherical drops, total oil volume and interfacial area satisfyTaking $10\mathrm{vol}\%$ to mean the oil fraction of the final emulsion, $V_o/(V_o+V_w)=0.10$ gives $V_o=V_w/9=1.111\times 10^{-4}{\mathrm{m}}^3$. Thus $A=333.3{\mathrm{m}}^2$ and $E_s=16.7\mathrm{J}$. This corresponds to about $2.65\times 10^{13}$ drops.

There is a wording convention worth making explicit: if the oil addition instead means $0.10$ times the initial water volume, then $V_o=100\mathrm{mL}$, $A=300{\mathrm{m}}^2$ and $E_s=15.0\mathrm{J}$. These differ only in the stated volume-fraction convention. The calculation uses the oil–water interface and neglects the container and air interfaces.

Iron is the negative anode and tin is the positive cathode of this spontaneous galvanic cell. electrons travel through the external circuit from iron to tin. An intact tin coating protects iron mainly by physically excluding the corrosive medium. If a defect exposes iron while the metals remain electrically connected, tin is not a sacrificial coating: the exposed iron can become the anodic corrosion site. A small exposed iron area coupled to a large cathodic tin area can be especially damaging; this distinction follows from the electrode-potential order rather than the appearance of the coating.

For a galvanic cell, subtract the standard anode reduction potential from the standard cathode reduction potential:A reduction potential is not multiplied by the number of transferred electrons.

The more negative standard reduction potential identifies iron as the species more readily oxidized. Combining $\mathrm{Fe}\to \mathrm{F}{\mathrm{e}}^{2+}+2{\mathrm{e}}^{-}$ with $\mathrm{S}{\mathrm{n}}^{2+}+2{\mathrm{e}}^{-}\to \mathrm{Sn}$ gives the spontaneous standard galvanic cell reactionThe electron balance is two per reaction, and the activities of the pure metals are unity.

Conventional semiconductor scanning capacitance microscopy is not a reliable dopant-contrast image of this specimen as drawn. Its usual $dC/dV$ signal relies on a tip–insulator–semiconductor capacitance whose depletion region changes under AC bias. An isolated conducting graphene sheet on thick insulating glass provides neither that conventional depletion geometry nor a controlled charge reservoir; simply earthing glass does not supply one. Capacitive or electrostatic contrast may be detectable with a suitably designed back electrode, a contacted flake and an appropriate capacitance-sensitive instrument, but that is an additional electrical device specification. Force-based surface topography remains possible independently of the specialized capacitance channel.

friction force microscopy is suitable. Contact scanning twists the AFM cantilever; the lateral photodetector difference measures friction and can distinguish the supported graphene and glass even if their heights are close. Acquire both trace and retrace and use half their lateral-signal difference to suppress slope-induced cross-talk. Keep the normal load low enough to avoid dragging or damaging the flake, control humidity and contamination, and calibrate torsional sensitivity if a quantitative coefficient of friction is required. An electrical connection is unnecessary.

quantitative nanomechanical mapping can image the flake without an electrical return path. Repeated AFM force curves provide surface topography, adhesion and deformation or effective elastic modulus channels. The step and differences in tip–surface interaction can distinguish graphene from glass. For an atomically thin supported sheet the contact response is strongly substrate dependent, so fitting a three-dimensional half-space contact model does not directly measure the intrinsic in-plane Young's modulus of free graphene. Calibrate the cantilever spring constant, deflection sensitivity and tip radius, use small loads, and compare bare substrate. The manufacturer's description specifies the force-curve origin of the mechanical channels.^[ref]

A steady-current image is not supported by the electrical connection drawn. tunnelling atomic force microscopy requires a conducting AFM probe, an applied bias voltage and a continuous return path from the contacted region to the current amplifier. graphene is conducting, but a flake on insulating glass is not electrically grounded merely because the other side or edge of the glass is earthed. Charging transients or displacement current are not its steady conductivity. Add a metal contact actually touching the graphene before using its current contrast; ordinary force-based surface topography can still be recorded without that contact. The manufacturer's nanostructure measurements explicitly distinguish objects connected to conducting pads from unconnected particles on the same insulating substrate.^[ref]