Materials Science · Hypothetical candidates and physical limits

Metallic-hydrogen candidates and persistence claims

Useful background: Energy, stability and environmental context; Measuring and checking material claims. Each worked case explains the quantities it uses.

Back to the chapter

After reading, you can

Distinguish in-pressure electronic evidence from ambient persistence and engineering function.
Explain why molecular identity, indirect electronic overlap and a direct optical edge are different questions.
Derive a bounded band-overlap example and a nonunique reflectance inference without identifying either toy model with real hydrogen.
Compare dated optical, electrical and pressure-calibration evidence while retaining the actual methods and limitations.
Separate sample coverage, retained response, reversibility under pressure and independently identified recovery after release.

Compression changes the electronic question

Hydrogen under extreme pressure can develop electronic states unlike ordinary molecular hydrogen. A 2020 Nature paper by Loubeyre, Occelli and Dumas reported infrared absorption evidence at about 80 K and near 425 GPa, interpreted by its authors as a probable transition to metallic hydrogen within the molecular solid. The wording and method matter.

Compression brings molecular units closer and changes orbital overlap, bonding and electronic bandwidths. It can reduce an electronic gap while recognizable H₂ units remain. Dissociation into an atomic structure is a different structural change. A molecular conductor and an atomic metal are therefore different candidate states; neither the word metallic nor a reflective appearance chooses between them.

Hydrogen’s light nuclei make zero-point motion important, even at low temperature. Electronic and nuclear approximations can change predicted phase boundaries and gaps. A static electronic calculation for one candidate lattice is not a complete quantum-nuclear, finite-temperature phase diagram. Atomistic models, assumptions and validation explains that representation issue; here the experimental observables and release claims are developed further.

The calculations below are original teaching examples with invented band parameters, optical constants and release records. They are not fits to raw hydrogen data, new electronic-structure calculations or demonstrations of a new phase. The cited experimental observations retain their dates, conditions and authors’ interpretations.

Electronic overlap can precede a strong direct optical edge

A crystal wave vector labels the spatial phase variation of an electronic state in a periodic lattice. A semiconductor has an occupied valence band and an empty conduction band in a simple zero-temperature picture. If the conduction minimum and valence maximum lie at different crystal wave vectors, their energy difference is an indirect gap. A photon carries very little momentum on the reciprocal-lattice scale, so a direct interband optical transition approximately keeps the electronic wave vector fixed. Indirect absorption can occur with momentum supplied by a phonon or another process, usually with a different strength and energy dependence.

Overlapping bands can produce electron and hole carriers without dissociating the molecules. A small overlap can give a semimetal with a low carrier density. Its resistance need not follow the temperature dependence of a good ordinary metal over every measured range: carrier population, scattering, contacts and disorder all matter. Finite conductance at one warm temperature alone could also arise from thermally activated carriers or another current path.

An invented local two-band model makes the distinction explicit. Let x be a dimensionless reduced wave-vector coordinate, not distance in the sample. Continue the labels valence-like and conduction-like branches from the insulating state, and set E_v(x) = −Ax² and E_c(x) = Δ + B(x − q)². A and B are positive energies; q is a dimensionless displacement between their extrema. Ignore other bands, electron–hole interactions, nuclear motion and changes of these curvatures. The coordinate range must include x = 0, x = q and the minimum-separation point.

The valence maximum is 0 at x = 0; the conduction minimum is Δ at x = q. Thus the indirect separation is Δ. At the same x, the branch separation is D(x) = Δ + Ax² + B(x − q)². Completing the square gives D(x) = Δ + (A + B)[x − Bq/(A + B)]² + ABq²/(A + B). Its minimum is D_min = Δ + ABq²/(A + B) at x* = Bq/(A + B). While this is positive, it is a nominal direct separation of these branches.

Take A = B = 1.00 eV, q = 1, and Δ(P) = 0.80 eV − (0.0100 eV/GPa) × (P − 300 GPa), only for 300–420 GPa. The indirect separation reaches zero at 380 GPa. At 400 GPa it is −0.200 eV, an overlap, while D_min is still +0.300 eV. The model therefore admits electronic overlap before this direct separation closes. It calculates neither conductivity nor an absorption coefficient: occupation, transition matrix elements, extra bands and scattering still enter those observables.

This smooth illustrative model is not a fit to an abrupt experimental absorption change or a hydrogen crystal structure. It only shows why an electrical onset and a direct optical-edge change need not mark the same event. Even a complete optical spectrum needs a response model; a finite spectral window that excludes an edge below 0.1 eV does not measure a gap exactly equal to zero.

Invented local band model; the pressure coordinate does not recalibrate any published experiment
P (GPa)	Indirect separation Δ (eV)	Minimum same-x separation (eV)	Model interpretation
300	0.800	1.300	Separated branches
340	0.400	0.900	Separated branches
380	0	0.500	Extrema first touch in energy
400	−0.200	0.300	Overlap with positive direct separation
420	−0.400	0.100	Overlap; small direct separation

Original invented local band model, not calculated or measured hydrogen bands. At model pressure 400 gigapascals, the valence-like branch is minus x squared electronvolts and the conduction-like branch is minus 0.200 plus the square of x minus one, with dimensionless reduced wave vector x from minus 0.5 to 1.5. Their extrema overlap by 0.200 electronvolt. At the same x equals 0.5, the branch energies are minus 0.250 and plus 0.050 electronvolt, a positive direct separation of 0.300 electronvolt. The lower plot evaluates the stated smooth invented pressure law from 300 to 420 gigapascals. The solid blue indirect separation crosses zero at 380 gigapascals; the dashed orange minimum same-x separation remains positive, reaching 0.100 electronvolt at 420 gigapascals. No actual transition pressure, conductivity, optical absorption or lattice identity is inferred.

Compare the observable and conditions before comparing the pressure number

A 2019 electrical/Raman study reported conduction beginning near 350–360 GPa below 200 K, interpreted as semimetallic molecular hydrogen. The electrical response and retained vibrational evidence supported a molecular band-overlap interpretation under those conditions. Resistance was measured through electrodes; conversion to resistivity also required specimen geometry. It was not a recovered ambient sample or a measurement of superconductivity.

The 2020 infrared report at about 80 K found a discontinuous change of the inferred direct gap near 425 GPa, from about 0.6 eV to below the roughly 0.1 eV accessible limit. The authors described a probable transition to a metallic state in the molecular solid. Infrared transmission returned on reducing pressure within the compressed-state measurements. That reversibility is a useful observation, not evidence that a metallic sample remained after complete release.

A separate 2020 first-principles analysis proposed that changes in the electronic bands of the same candidate molecular structure could explain the optical change without requiring a structural transition. It also acknowledged electronic-approximation and omitted nuclear-quantum effects. That is an alternative interpretation to test, not experimental identification of the calculated structure. Agreement with an optical trend does not uniquely choose a lattice.

These reports need not all be reduced to competing numerical estimates of one universal transition pressure. They probe different observables and conditions, and their pressure scales must be examined. A real disagreement remains possible; resolving it requires matched response definitions, calibration and material states, rather than choosing whichever report has the largest pressure or brightest image.

Selected dated claims and their remaining boundaries; no settled universal phase diagram is asserted
Record	Actual main observable	Reported interpretation	Not established by that record
2019; roughly 350–360 GPa, below 200 K	Electrical response with Raman evidence	Semimetallic molecular hydrogen	Atomic lattice, superconductivity or ambient recovery
2020; near 425 GPa, about 80 K	Infrared edge and vibrational response	Probable metallization within the molecular solid	Unique structure or persistent ambient metal
2020 theoretical alternative	Calculated bands and vibrational trends	Same candidate molecular phase can explain optical change	Direct structural identification or experimental lifetime

Reflectance is a boundary response with a nonunique inversion

For an instructive optical boundary, assume a plane wave at normal incidence from a lossless window of real refractive index n_d into a homogeneous, isotropic, nonmagnetic, semi-infinite sample. Let its complex refractive index be N = n + iκ with κ ≥ 0. Here i² = −1, ω is angular frequency, z is depth and c is the vacuum speed of light; with time dependence exp(−iωt), propagation exp(iNωz/c) decays into the sample when κ is positive. This excludes finite-film interference, multiple reflected interfaces, roughness, scattering and anisotropy.

Let E_i, E_r and E_t be the incident, reflected and transmitted tangential electric-field amplitudes. Continuity of tangential electric and magnetic fields gives E_i + E_r = E_t and n_d(E_i − E_r) = N E_t for this nonmagnetic boundary. Therefore the reflected electric-field amplitude ratio is r = (n_d − N)/(n_d + N). The reflected power fraction is R = |r|² = [(n − n_d)² + κ²]/[(n + n_d)² + κ²]. One reflectance at one photon energy supplies one relation for two optical constants, before any additional dispersion model is chosen.

For invented n_d = 2.40 and R = 0.640, both (n, κ) = (21.60, 0) and (2.40, 6.40) satisfy the equation. One is lossless at that frequency; the other attenuates light inside it. Neither pair is asserted to describe hydrogen or the actual high-pressure window. These are point values, not complete causal spectra; extra frequency-dependent information would be needed to specify a full response.

The distinction matters because optical absorption at finite frequency is not itself proof of a free-carrier response at zero frequency. Interband excitations can absorb in a nonmetal. A Drude description instead models mobile-carrier acceleration and relaxation; its conductivity has the form σ(ω) = n_e e²τ/[m*(1 − iωτ)], with carrier number density n_e, carrier charge magnitude e, effective mass m* and relaxation time τ. Its extrapolated σ(0) depends on those assumptions and any other response/background terms, not on shiny appearance alone.

At a known vacuum wavelength λ₀, intensity in a uniform medium decays with absorption coefficient α = 4πκ/λ₀. An optical analysis also needs the window’s own attenuation, reflection/background corrections, sample geometry and spectral range. For a closed passive power budget at common reference planes, R + T + A_abs = 1. If scattered light or an unmeasured path is omitted, loss of collected transmission cannot simply be assigned to absorption in hydrogen.

The pressure coordinate is inferred through a calibrated stressed solid

In a diamond-anvil experiment the applied load is not a direct reading of a uniform sample pressure. The stressed diamond produces a Raman band whose high-wavenumber edge can be related empirically to a pressure reference. Gradients, deviatoric stress, sampling location and the calibration model matter. A change of pressure scale moves a result along the horizontal axis; it does not change the recorded photon counts or prove a new phase.

A 2023 calibration study correlated diamond-edge Raman shifts with gold diffraction and an equation of state over its measured range. Its fitted pressure scale differed substantially from extrapolations of earlier scales at the highest pressures. This is specific primary evidence that calibration can affect comparisons among dense-hydrogen reports. The source title uses “universal”; that word is not a guarantee for every stress field, geometry, temperature or later experiment.

To illustrate use of the published functional form, P = A r + B r², take r = (ω_edge − ω₀)/ω₀, ω₀ = 1332.5 cm⁻¹, A = 517 GPa and B = 764 GPa from that study. For a hypothetical edge shift with r = 0.500, P = 449.5 GPa. This evaluates the stated empirical scale; it is not a new measurement or a retrospective correction of the 2020 record.

To compare studies, retain the measured edge definition and position, calibration and its range, uncertainty, temperature and sampled region. A random uncertainty around one scale does not automatically cover a systematic difference between scales. Do not apply one fixed percentage correction to every published hydrogen pressure: the mapping is nonlinear and the appropriate original observable and stress context are needed.

Electronic evidence does not determine molecular or chemical identity alone

A molecular vibron probes intramolecular vibration and can support persistence of molecular units in the sampled region. Its frequency, width and intensity have different meanings. Screening, optical attenuation, selection rules and background can suppress a measurable line. Losing the line does not by itself prove that every molecule dissociated. Conversely, observing a vibron can coexist with electronic carriers in a molecular semimetal.

A structural assignment should compare alternative molecular and atomic arrangements using observables that distinguish them. A spectrum consistent with one calculated lattice need not exclude all others. Diffraction or complementary structural constraints may help, but their sensitivity and actual accessible region remain part of the evidence. A fitted optical carrier model is not an independent crystal-structure measurement.

Keep elemental hydrogen separate from a hydrogen-rich compound. Adding another element changes composition, bonding and the candidate phase; an unusual response of a hydride is not an observation of the same response in pure hydrogen. Electrode, gasket, window or reaction products also require their own identification when they can contribute to the signal or current path. This is why traceable specimen identity and matched backgrounds accompany optical or transport evidence.

One claim does not establish all desired properties

Optical or transport evidence under pressure is not automatically proof of a specific crystal structure, superconducting state, recoverable ambient sample or usable energy technology. Evaluate each claim through its actual observables, calibration and independent reproduction. Later evidence should be assessed on its own terms rather than inferred from this example.

A high-pressure state can be an equilibrium minimum at one pressure and cease to be a local minimum on release. If a local minimum persists but a lower state exists, a barrier may permit metastability over some interval. Mechanical stability against small displacements and the rate of a chemical or structural transformation are different tests. Metastability and persistence and Decompression, metastability and retained properties develop that distinction.

A first-principles study of proposed atomic hydrogen structures examined how their relative energies and vibrational instabilities changed during decompression. Its candidate set and approximation choices limit the conclusion; a dynamically stable candidate at a finite pressure does not supply an ambient lifetime. Locating a stationary atomic structure is weaker than showing that it resists all relevant allowed perturbations.

A published 2023 metastability report describes reducing the load at 5 K after a reflective high-pressure state was interpreted as metallic. Its published abstract states that the sample apparently transformed and escaped as the sample hole closed, that the exact metastability pressure could not be determined, and that zero-pressure metastability was not observed. Preserve that negative outcome and its sample-loss/pressure limits. It neither demonstrates recovered material nor proves that every hypothetical metallic structure under every condition is impossible.

A release signal can mix retained state and remaining specimen area

An invented reduced-pressure observation shows why tracking the same specimen matters. Imagine that independent identification has already established reference states M and I at 100 GPa and 5 K. The assumed references have reflectances R_M = 0.640 and R_I = 0.160 at one photon energy; an exposed background has R_b = 0.040. These hypothetical reference states and constants are not a claim that an ambient or reduced-pressure hydrogen metal has actually been prepared.

Let c be the fraction of the illuminated area still covered by specimen, and f the fraction of that covered area in M. Assume uniform illumination, patches large relative to wavelength, incoherent addition of their reflected powers, no multiple scattering and unchanged reference responses. Then R_obs = c[fR_M + (1 − f)R_I] + (1 − c)R_b. This area-weighted model would not apply unchanged to subwavelength mixtures, coherent interference or unknown reference constants.

For a hypothetical R_obs = 0.280, c = 1 gives f = 0.250, whereas c = 0.400 gives f = 1. Both have the same measured reflectance. The latter has lost 60% of its illuminated specimen coverage but all of the remaining covered area is in the stipulated reference state. A single optical mean cannot separate coverage from retained fraction.

An independent coverage map can break this particular ambiguity. At known c = 0.800, subtract the background contribution (0.200)(0.040) = 0.008, divide the remainder by 0.800 and obtain covered-area reflectance 0.340. Then f = (0.340 − 0.160)/(0.640 − 0.160) = 0.375. Even that conditional fraction relies on the assumed reference identities and mixing model. It is not, by itself, a new structural measurement or proof of metallicity.

A record at 100 GPa is still far from ambient pressure. State the pressure, temperature, retained identity, functional response, remaining material and observation duration separately. A pressure cycle with a reversible optical change establishes a path-dependent response within the investigated interval; it does not establish that a metallic sample was recovered at the end.

Two original invented inverse examples, not measurements of hydrogen. The upper plot gives all passive pointwise optical pairs on the upper semicircle with centre n equals 10.933333 and radius 10.666667, for normal-incidence reflectance 0.640 from a lossless window of index 2.40. Pair P has n equals 21.60 and extinction coefficient zero; pair Q has n equals 2.40 and extinction coefficient 6.40. Both yield the same reflected power. They do not specify a full causal optical spectrum or dc conduction. The lower plot uses separate hypothetical identified reference states at 100 gigapascals and 5 kelvin, with reflectances 0.640 and 0.160 and background 0.040. At observed mean reflectance 0.280, the retained reference-state fraction f equals 0.5 divided by specimen coverage c minus 0.25. The allowed coverage interval is 0.400 to 1; marked pairs are (0.400,1), (0.800,0.375), and (1,0.250). A second coverage measurement is needed. These are reduced-pressure bookkeeping examples, not ambient recovered material.

Define the recovery claim before interpreting a finite record

For a meaningful persistence comparison, specify the desired state and function, the pressure–temperature path and the final environment. Identify the remaining specimen at the endpoint and examine the relevant electronic response there over a stated interval. Pair before/after regions and suitable backgrounds, and retain detection limits. Otherwise an apparent loss of a state can be confused with loss of the specimen, or a remaining bright region with recovered bulk material.

A change during decompression can reveal where the starting state is no longer retained under that protocol, if the state and pressure remain measurable. A closed or vanished sample chamber may prevent locating that change precisely. No observation after a loss of access should be silently filled in with a theoretical transition pressure. Likewise, missing a spectral edge below an instrument window is a bound on what was resolved, not a measurement of an exactly zero gap.

Metallicity, superconductivity, metastability and engineering function remain different claims. Superconductivity requires its own electrical and magnetic evidence and controls; an ordinary conducting or reflective state does not imply it. A hypothetical application additionally needs reproducible quantities, retention, safe integration and performance under its intended conditions. None of the dated optical, electrical or release records described here supplies that complete chain of evidence.

Worked examples

Work through the case

An experiment supports a metallic response while pressure is maintained. Has ambient metastability been demonstrated?

The observation concerns the compressed state.
A release experiment must establish retained identity and duration.
Properties of the retained state require their own measurements.
Reversing an optical change while the sample is still compressed is not the same endpoint as identifying and testing a recovered ambient specimen.

No. Ambient persistence is a separate testable claim.

Find electronic overlap without assigning an atomic structure

Use the invented A = B = 1 eV, q = 1 model at 400 GPa. Locate the two extrema and minimum same-x separation. Would an electrical response alongside a positive direct separation be contradictory? Then change only q to 0.80 at the same pressure.

At 400 GPa, Δ = 0.80 − 0.0100(400 − 300) = −0.200 eV. The valence-like maximum is 0 at x = 0 and the conduction-like minimum is −0.200 eV at x = 1. Their energy ranges overlap.
Complete the square: D(x) = −0.200 + 2(x − 0.5)² + 0.500 eV. Thus x* = 0.500 and D_min = 0.300 eV. At that x the two branch energies are −0.250 and +0.050 eV.
The overlap permits a semimetal interpretation within a suitable occupied-band model, while direct transitions can retain a positive separation. Electrical conduction and a positive direct separation therefore need not conflict. This local model computes neither carrier mobility nor the actual optical edge.
With q = 0.80, the extrema are now displaced by 0.80 in x and x* = 0.400. The offset ABq²/(A + B) is 0.320 eV, so D_min = 0.120 eV. The indirect overlap remains −0.200 eV. A changed momentum separation changes the optical comparison without changing that overlap.

The original model overlaps by 0.200 eV while retaining a 0.300 eV same-x separation. Changing q to 0.80 reduces the latter to 0.120 eV. Neither case identifies hydrogen’s molecular or atomic structure.

Invert one reflected-power measurement without inventing uniqueness

At one frequency, the invented window has n_d = 2.40 and R = 0.640. Verify the two optical-constant pairs in the text. For the absorbing pair at vacuum wavelength 2.00 μm, calculate α and the intensity attenuation length. Does this establish dc metallic conduction?

For n = 21.60, κ = 0, R = (19.20/24.00)² = 0.640. The high reflected fraction is compatible with no absorption at this frequency in this pointwise model.
For n = 2.40, κ = 6.40, R = 6.40²/[(4.80)² + 6.40²] = 40.96/64.00 = 0.640. The same measured R permits a different complex index.
For the second pair, α = 4πκ/λ₀ = 4π(6.40)/(2.00 × 10⁻⁶) = 4.02124 × 10⁷ m⁻¹. The intensity attenuation length 1/α = 2.48680 × 10⁻⁸ m, about 24.9 nm. This is not the electric-field amplitude decay length, which is twice as large.
Finite-frequency absorption is compatible with several electronic mechanisms. These two point values do not specify a frequency-dependent Drude response, carrier density or σ(0). Nor are the invented constants measured high-pressure hydrogen data.

Both pairs give R = 0.640. The absorbing pair has α ≈ 4.02 × 10⁷ m⁻¹ and intensity attenuation length ≈ 24.9 nm. No dc-metal or structural identification follows from that one optical point.

Evaluate a calibration without relabelling a historical experiment

For the selected 2023 empirical equation, take a hypothetical measured edge with r = 0.500. Find its Raman edge position and the central pressure. If another scale assigns a different pressure to that same edge, did the optical observation change?

The dimensionless r equals (ω_edge − ω₀)/ω₀. Therefore ω_edge = (1 + 0.500)(1332.5) = 1998.75 cm⁻¹.
Evaluate P = 517(0.500) + 764(0.500)² = 258.5 + 191.0 = 449.5 GPa. Coefficients carry GPa, so the result is a pressure.
The edge position is the same observation under a different calibration. A new mapping changes the inferred pressure coordinate and its uncertainty, not the recorded spectrum. Re-examining a historical sample would require its actual original edge and applicable stress context.
The displayed calculation uses central coefficients only. The source’s fitted-coefficient uncertainties, their covariance, spectral-edge uncertainty and equation-of-state/stress contributions are not replaced by the last decimal place of 449.5.

The hypothetical edge is 1998.75 cm⁻¹ and the selected scale gives 449.5 GPa. This is a scale evaluation, not an independent pressure measurement or a recalibration of the 2020 result.

Use a second observable to distinguish two release interpretations

For the invented 100 GPa, 5 K references, R_obs = 0.280. Verify the c = 1 and c = 0.400 interpretations, then use an independently supplied coverage c = 0.800 to find f. Could the last result establish ambient recovery?

At c = 1, the background vanishes. Solve 0.280 = 0.160 + f(0.640 − 0.160), giving f = 0.250.
At c = 0.400 and f = 1, R_obs = 0.400(0.640) + 0.600(0.040) = 0.256 + 0.024 = 0.280. The optical mean cannot distinguish this coverage loss from the first interpretation.
At c = 0.800, the covered part has mean reflectance [0.280 − (0.200)(0.040)]/0.800 = 0.340. Then f = (0.340 − 0.160)/0.480 = 0.375.
The independent coverage observation removes one unknown only within the supplied mixing and reference model. The pressure is still 100 GPa, identity was stipulated rather than discovered here, and no ambient-state time record is supplied.

The same mean permits (c, f) = (1, 0.250) or (0.400, 1). Supplied c = 0.800 gives f = 0.375. None of these invented reduced-pressure results establishes ambient recovery.

Try the reasoning

Does metallic imply superconducting at every temperature?
Show the answer and reasoning
No.
They are different physical states.
Why preserve “probable” in discussing the cited study?
Show the answer and reasoning
It reflects the authors’ evidential interpretation.
A source’s scope should not be strengthened in retelling.
In the original invented band model, find Δ and D_min at 390 GPa. If q alone is changed to 0.60 with A = B = 1 eV, what becomes the minimum same-x separation and where does it occur?
Show the answer and reasoning
At 390 GPa, Δ = −0.100 eV and D_min = 0.400 eV for q = 1. For q = 0.60, x* = 0.300 and D_min = −0.100 + 0.180 = 0.080 eV.
The indirect overlap is unchanged by q in this model because the extrema keep the same energies. The direct separation depends on their momentum displacement. No absorption strength or real transition pressure is calculated.
A hypothetical passive optical measurement at common reference planes has R = 0.640 and T = 0.100, with scattering and every other channel excluded. Find A_abs. Does replacing the measured T by a detector limit T < 0.010 establish zero transmission or a zero electronic gap?
Show the answer and reasoning
For the first record, A_abs = 0.260. For the limited record, 0 ≤ T < 0.010 implies 0.350 < A_abs ≤ 0.360 in this closed model, not T = 0 or a zero gap.
Use conservation of incident power and retain the strict detector bound. A finite spectral/power limit is not an exact zero; if uncollected scattering or window losses exist, the assumed assignment to sample absorption must be reconsidered.
Use the invented absorbing optical pair at λ₀ = 2.00 μm. Over thickness d = 50.0 nm, what fraction of the intensity already inside the uniform medium remains after propagation? Does this fraction include entrance reflection?
Show the answer and reasoning
exp(−αd) = exp[−(4.02124 × 10⁷)(50.0 × 10⁻⁹)] ≈ 0.1339. It excludes entrance reflection.
The intensity propagation factor is separate from the boundary reflectance and any exit interface or multiple reflection. The electric-field amplitude factor is its square root. These are stipulated optical constants, not hydrogen measurements.
Keep the invented release references, but now R_obs = 0.310 and independently known c = 0.750. Find the covered-area reflectance and f. What fails if c is inferred from the same single R instead of measured independently?
Show the answer and reasoning
The covered-area reflectance is [0.310 − 0.250(0.040)]/0.750 = 0.400. Thus f = (0.400 − 0.160)/0.480 = 0.500. One R alone leaves two unknowns.
A second quantity is informative only when it supplies distinct evidence and the mixing/reference assumptions remain valid. Reusing one reflectance to choose both coverage and fraction does not add a constraint.
For the selected P = Ar + Br² equation, compare the central pressures at r = 0.500 and r = 0.510 with A = 517 GPa and B = 764 GPa. Is their difference a complete pressure uncertainty?
Show the answer and reasoning
P(0.500) = 449.5 GPa; P(0.510) = 462.3864 GPa; the change is 12.8864 GPa. It is a sensitivity comparison, not a complete uncertainty.
The nonlinear relation gives ΔP = (A + 2Br)Δr + B(Δr)² here. Coefficient covariance, the reference equation of state, stress and edge-selection effects still need their own evaluation.
A compressed sample shows a molecular vibron and finite electrical conductance. After full release, a dark spot remains but its composition and electrical response are not measured. Which claims remain separate?
Show the answer and reasoning
The in-pressure evidence can support a conducting molecular state under its actual controls. The remaining spot does not establish a retained atomic phase, ambient metal, superconductivity or useful persistence.
Molecular identity can coexist with carriers. A visible residue is not a phase or composition determination, and the electronic response was measured under a different pressure. A retained endpoint needs appropriate identity, response, conditions and time evidence.

What to carry forward

Attach pressure and method to the claim.
Assess metallicity, superconductivity and retention separately.
Indirect band overlap, a direct optical edge and molecular dissociation need not coincide.
A pressure number and a fitted optical constant retain their calibration, boundary and model assumptions.
Reversibility within a compressed-state cycle and recovery of a persistent ambient specimen are different endpoints.

References

Teaching resources and research supporting this lesson’s explanations and examples.