Experiments
Registered experiments, their write-up status, stored
datasets, and pending derivations.
| experiment chapters |
40 |
| experiments in them |
110 |
| written up under all five headings |
98 |
| still owing at least one heading |
12 |
| entries in the List of Experiments |
110 |
with a dataset stored in docs/library/data |
28 |
| primary sources cited across them |
1134 |
| derivations still pending |
87 |
Jump to
the 87 pending derivations
or the 12
missing write-ups, both listed in full below the tables.
complete
content with pending derivations
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Classical Mechanics
Special Relativity
General Relativity and Cosmology
Classical Thermodynamics
Classical Field Theory and Electrodynamics
The Transition to Quantum Physics
Quantum Mechanics
Relativistic Quantum Mechanics
Quantum Field Theory and the Standard Model
Statistical Mechanics
Frontiers and Open Problems
Pending derivations
Open derivation notes from experiment chapters, in book
order.
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Experiment: The Mechanical Equivalent of Heat: all derivations of this chapter are pending.
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The Chapman–Enskog transport coefficients: that the first-order solution of the Boltzmann transport equation about local equilibrium fixes the numerical coefficients left undetermined by the mean-free-path argument, and in particular gives the ratio of thermal conductivity to viscosity times specific heat as 5/2 for a monatomic gas, against the value 1 of the elementary estimate.
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The residual internal field of a paramagnetic salt: that the dipolar sum over the lattice of magnetic ions, together with the crystal-field splitting of the ground multiplet, fixes the ordering temperature and hence the floor under a single demagnetization stage, so that the cooling ratio is a computable property of the salt rather than a fitted parameter.
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The spectroscopic entropy of a gas: that the translational, rotational, vibrational and electronic partition functions, evaluated from band-spectrum constants and molecular masses, give the absolute molar entropy against which the calorimetric integration is compared, together with the symmetry number and the nuclear-spin terms that cancel between the two routes.
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Experiments: Electromagnetism: all derivations of this chapter are pending.
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The Coulomb exponent: derive the interior field of a uniformly charged spherical shell for a force law with exponent \(2+\delta\), show that it vanishes identically only when \(\delta\) is zero, and give the conversion of a measured null voltage between concentric conductors into a bound on \(\delta\). Then relate \(\delta\) to a photon rest mass through the Yukawa form that a massive vector field gives in place of the Coulomb potential.
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The ratio of the units: show that the electrostatic and electromagnetic definitions of the unit of charge differ by a factor with the dimensions of a speed, equal in SI terms to the reciprocal square root of the product of the electric and magnetic constants, and hence that this measurement determines exactly the combination that reappears as the propagation speed in the vacuum wave equation. The error budget of the original determination, and its agreement with the contemporary optical values, are to be reconstructed here.
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Experiment: Wave Optics: all derivations of this chapter are pending.
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The diffraction limit: derive the chromatic resolving power of a grating from the angular width of an order — the wavelength divided by the order times the number of rulings — using the \(N\)-aperture sum and the convention that two lines are resolved when the maximum of one falls on the first zero of the other. Then derive the imaging bound from the requirement that at least two diffracted orders from a periodic object be collected by the objective, which gives the smallest resolvable period as the wavelength divided by twice the numerical aperture, and state the hypotheses — linearity in the illumination, far-field collection — on which the bound depends.
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Diffraction efficiency of a hologram: what fraction of the reconstructing beam is delivered into the reconstruction, derived for a thin absorption record and a thin phase record by expanding the transmittance of a sinusoidal grating, and for a thick phase record by the coupled-wave treatment in which two waves exchange energy under the Bragg condition. What has to be established is that only a volume record can send almost all of the incident light into a single order — which is what makes a white-light reflection hologram bright enough to look at — while the thin records are bounded well below that.
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Single-quantum interference: the derivation is quantum and belongs to the later parts. What must be shown here is negative and can be made precise — that no classical field theory reproduces the combination of a single localized detection with a fringe pattern of full visibility, since the second-order correlation at zero delay is bounded below by unity for any classical field, while the measured value for a heralded single photon falls well below it. The wave account of this chapter must therefore be read as the high-occupancy limit of something else.
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Experiment: Franck–Hertz: all derivations and data analysis of this chapter are pending.
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Why the glow layers sit where they do: an electron starting from rest at the cathode reaches the excitation energy after falling through a fixed potential difference, so the luminous shells mark equal potential intervals, and their migration towards the cathode as the voltage is raised follows from the field distribution in the tube. The dependence of the effective spacing on current and geometry requires in addition the branching among the accessible neon levels and the collision cross-section of each.
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The effective spacing: the excitation cross-sections of the competing mercury levels as functions of electron energy, and the weighted mean interval they produce in a tube of given vapour density and geometry; and the mechanism of the drift with maximum number, in which electrons surviving several accelerating stages sample a different mixture of levels from those excited near the first drop.
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Experiment: The Blackbody Spectrum Measured: what is pending is the tabulation of the measured radiance against wavelength and temperature for each experiment, in SI with its uncertainty budget, together with the residuals against the Wien and the Planck distributions. The phenomena stated below each carry their derivation inline.
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Experiment: The Photoelectric Effect and Compton Scattering: what is pending is the tabulation of the measured numbers for each experiment, in SI with its uncertainty budget — stopping potential against frequency for each emitting metal, scattered wavelength against scattering angle, coincidence and singles counts against resolving time, and the gated detection probabilities of the beamsplitter experiment. The phenomena stated below carry their derivations inline except where a pending derivation is marked against them individually.
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The operational definition of the stopping potential. The measured current does not fall to zero at a sharp potential but approaches it along a tail, because the emitted electrons have a distribution of energies rather than a single maximum, because electrons originate at a range of depths below the surface, and because the collector is not equipotential. What is pending is the derivation of the shape of that current–voltage curve near its foot from the energy distribution of the photoelectrons, and the demonstration that the extrapolation Millikan used recovers the true intercept without bias — the point being that the linearity of the plotted line is a statement about a quantity defined by an extrapolation, and that extrapolation must be shown not to have manufactured it.
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The angular correlation measured by the cloud-chamber experiment. What is pending is the direction of the recoil electron as a function of the scattering angle of the quantum, obtained from the same conservation laws that give the wavelength shift, together with the angular spread introduced by the initial motion of the struck electron and by multiple scattering of the recoil track in the chamber gas — the latter being what sets the accuracy with which the predicted correlation can be confirmed, and hence what the measurement is really comparing.
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Experiment: Matter Waves Observed: what is pending is the apparatus and procedure of each measurement and the tabulation of its data — scattered intensity against angle and accelerating voltage, ring radii against voltage, diffraction angle against source temperature, fringe visibility against count, neutron intensity against tilt angle, and molecular fringe contrast against internal temperature and residual gas — each in SI with its uncertainty budget. The phenomena stated below each carry their derivation inline.
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The quantitative decoherence rates: the thermal photon emission spectrum of a hot molecule and the resulting product of which-path factors as a function of internal temperature, giving the predicted visibility curve; and the collisional case, in which the localization length follows from the momentum transfer distribution of the residual gas at a given pressure and temperature, giving the predicted decay of visibility with pressure. Both are stated qualitatively in the derivation above from the single-photon overlap; what must be supplied is the master equation whose decoherence rate they are, together with the statement of which of its assumptions the data actually test.
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X-ray spectra and the charge of the nucleus: the classical account of the emission itself, namely the bremsstrahlung continuum radiated by cathode-ray electrons decelerated in the anticathode, and its short-wavelength cut-off.
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Characteristic X-radiation: the quantum account, in which the primary beam ejects an electron from an inner shell and the radiation is emitted when an outer electron fills the vacancy, so that the excitation threshold is the binding energy of the shell and the monotone hardening along the table is the growth of that binding energy with nuclear charge.
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The X-ray term system: the derivation of the K, L and M level multiplicities from the quantum numbers of the many-electron atom, with the L and M splittings as the combined effect of screening and of spin-orbit coupling, and the identification of an absorption edge with a one-electron binding energy.
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Force on a magnetic dipole in an inhomogeneous field
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Quantum derivation of the two-valued splitting from spin-1/2 (to Angular Momentum and Spin)
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Sequential SG probabilities from the measurement postulates
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Experiment: Bell Tests: all derivations of this chapter are pending.
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The detection-efficiency threshold: the least heralding efficiency at which the observed coincidence and singles rates cannot be reproduced by any local model free to decide which pairs go undetected, worked out both for the CHSH inequality with a maximally entangled state and for the Eberhard inequality with a partially entangled one, the latter threshold being the lower and being the reason the photonic experiments deliberately used a non-maximally entangled state
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Experiment: Precision Spectroscopy and Atomic Clocks: all derivations of this chapter are pending.
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Magnitude of the interband tunnel current: the transmission of a carrier across the forbidden gap of a uniformly biased semiconductor, the imaginary wavevector inside the gap obtained by continuing the two-band dispersion relation, and the resulting exponential dependence of the peak current on the gap, the effective mass and the junction field
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The tunnel current between a normal metal and a superconductor: the elastic-tunnelling current as the convolution of the two densities of states with the difference of the two Fermi functions [Bardeen:1961], the BCS quasiparticle density of states with its square-root edge at the gap, and the consequence that the differential conductance at low temperature is proportional to that density of states evaluated at the bias energy
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The tunnel current as a probe of the local density of states: the transfer-Hamiltonian expression for the current between two weakly coupled electrodes in terms of the overlap of their surface wavefunctions [Bardeen:1961], and its reduction, for a tip modelled by a single spherically symmetric orbital, to the local density of states of the sample at the position of the tip and at the bias energy [Tersoff:1985]
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Experiment: The Positron: all derivations of this chapter are pending.
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The identification of the penetrating cosmic-ray component as a heavy lepton rather than as the quantum of the nuclear force: its anomalously weak absorption in nuclei against the strong coupling a nuclear-force carrier must have, and the three-body shape of its decay electron spectrum. The muon's own phenomenology is taken up in the cosmic-ray chapter of Part XII.
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The CPT theorem — that any local, Lorentz-invariant quantum field theory with a Hermitian Hamiltonian and the normal spin–statistics connection is invariant under the combined operation of charge conjugation, parity and time reversal — together with the equalities of mass, lifetime and magnetic moment that follow from it, and the reasons a null result here bounds rather than confirms.
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Experiment: Neutrino Helicity: all derivations of this chapter are pending.
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The Michel spectrum: the most general Lorentz-invariant four-fermion matrix element for muon decay, its reduction to four measurable parameters in the decay distribution, and the values those parameters take when the current is the purely left-handed one. Deriving it also shows which combinations of couplings the experiment cannot separate, which is what makes the bounds on right-handed admixtures the shape they are.
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The parity-violating asymmetry in the scattering of longitudinally polarized electrons, as the interference between the photon-exchange and neutral-current amplitudes, and its expression in terms of the weak mixing angle; together with the analogous expression for the parity-non-conserving amplitude in a heavy atom, where the nuclear weak charge and the atomic structure factorize. Both belong to the electroweak chapter of Part XII.
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The general left–right parametrization of the charged current, the observables in beta and muon decay that are first order in the right-handed admixture as against those that are second order, and the translation of a null asymmetry into a bound on the mixing angle and on the mass of an additional charged vector boson. The distinction between the two orders is what determines how the sensitivity scales, and it must be stated with the bounds.
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Doubly differential muon-decay rate from the \(V-A\) four-fermion matrix element, i.e. the angular factor \({[}(3-2x)+P_{\mu}(2x-1)\cos\theta{]}\) that is quoted rather than computed here; the energy spectrum alone is parametrized in the weak-interactions chapter, and the integration of the angular factor is carried out in this chapter. Belongs in the long-proofs appendix
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Atomic many-body calculation converting the measured caesium amplitude into a nuclear weak charge: the relativistic coupled-cluster evaluation of the valence electron density at the nucleus and its error budget. Belongs in the long-proofs appendix; the mixing of opposite-parity states and the \({Z}\)-cubed scaling are derived in this chapter
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Coefficients of the deep-inelastic parity-violating asymmetry on an isoscalar target: the combinations \(1-\tfrac{20}{9}\sin^{2}\theta_{W}\) and \(1-4\sin^{2}\theta_{W}\) follow from the electron and quark neutral-current couplings \(C_{1q}\), \(C_{2q}\) contracted with the quark-parton structure functions of a deuteron, and are quoted rather than derived here. Belongs in the long-proofs appendix; the couplings themselves are written down in this chapter for the proton case, and the deep-inelastic kinematics live in the deep-inelastic chapter
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One-loop electroweak corrections to the weak charge of the proton: the \(\rho\) factor, the charge-radius term, and the \(WW\), \(ZZ\) and \(\gamma Z\) box graphs that together shift the tree value \(1-4\sin^{2}\theta_{W}=0.047\) to the Standard Model prediction near \(0.071\). They are a third of the predicted value, so the comparison with experiment does not exist without them. Belongs in the long-proofs appendix
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The short-distance box-diagram computation of the \({[}\Delta S=2{]}\) mixing amplitude, giving the mass difference and epsilon in terms of the CKM elements and the quark masses — the step that converts the measured epsilon into a constraint on the unitarity triangle. It is an appendix-scale calculation requiring the loop functions of the box with internal charm and top, and it belongs in Appendix A rather than in an experiment chapter.
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The \(U\)-spin relation between the direct \(CP\) asymmetries of \({[}B^{0}\to K^{+}\pi^{-}{]}\) and \({[}B_{s}^{0}\to K^{-}\pi^{+}{]}\) is stated here in outline only. Owed: the explicit tree-plus-penguin decomposition, the action of the \(d \leftrightarrow s\) interchange on each amplitude, and the width weighting that turns the amplitude statement into the ratio \(-1\), together with the size of the \(U\)-spin-breaking corrections. This belongs in Appendix A. No key in the bibliography covers the \(U\)-spin literature, so the claim carries no citation.
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The three-loop, chirally suppressed Cabibbo–Kobayashi–Maskawa contribution to the neutron electric dipole moment is quoted here as an order of magnitude and is not derived. Owed: the argument that the one- and two-loop quark diagrams cancel by the Glashow–Iliopoulos–Maiani mechanism because the rephasing invariant needs all six quark masses, and the resulting estimate of order \(3\times 10^{-53}\,\mathrm{C}\,\mathrm{m}\). This belongs in Appendix A. The bibliography carries no key for the electric-dipole calculation, so the estimate is uncited.
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The invariance theorem for a tilted and elliptical Penning trap: the general Brown–Gabrielse result, that the sum of squares of the three measured mode frequencies equals the squared free-space cyclotron frequency independently of the tilt angle and the ellipticity parameter. Belongs in Appendix A beside the ideal-trap identity proved in this chapter.
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The dispersive representation of the hadronic vacuum-polarization contribution to a lepton anomaly: analyticity of the photon self-energy, the optical theorem relating its discontinuity to the total hadronic cross section, and the derivation of the kernel \({[}K(s){]}\) from the one-loop vertex with a dressed photon propagator. Belongs in Appendix A.
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The coefficient of the scalar-mass logarithm in the radiative correction \(\Delta r\), written parametrically in this subsection, together with the constant term that accompanies it — which is not a refinement at the measured mass but comparable to the logarithm itself. Obtaining both requires the one-loop \(W\) and \(Z\) self-energies with the scalar circulating, renormalized on shell. The electroweak chapter already records the same logarithm as owed, in the paragraph following its screening-theorem discussion; the two belong together in Appendix A. Nothing in this chapter is evaluated numerically from the coefficient.
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The Standard Model production cross sections quoted in this section. Obtaining them requires the partonic amplitudes — above all the top-quark triangle that drives gluon fusion, which has no tree-level counterpart — convolved with the measured parton distributions and corrected to next-to-next-to-leading order in the strong coupling, where the corrections are large (a \(K\) factor near two for gluon fusion). The loop amplitude and the resulting cross sections are already recorded as owed in the electroweak chapter's pending derivation of the loop-induced scalar vertices; the convolution and the perturbative corrections belong with them in Appendix A. What is derived in this chapter from those inputs is everything downstream of them: the produced event counts, the channel comparison, the mass resolution and the significance.
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The azimuthal distributions that separate \(0^{+}\) from \(0^{-}\) and from the spin-\(2\) hypotheses in \(H\to ZZ^{*}\to4\ell\). What is owed is the decay amplitude for each hypothesis — the scalar contraction of the two boson polarizations, the Levi-Civita contraction that a pseudoscalar is restricted to, and the tensor couplings — projected onto the two lepton pairs and integrated to give the distribution in the angle between their decay planes, exhibiting the \(\cos\Phi\) term the scalar produces, its absence for the pseudoscalar, and the \(\cos2\Phi\) term of opposite sign that the pseudoscalar carries. It belongs in Appendix A beside the vector-boson vertices of the electroweak chapter, whose loop-induced counterparts are already recorded as owed there.
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The double-production cross section \(\sigma(gg\to HH)\) as a function of \(\kappa_{\lambda}\): the interference of the top-loop triangle carrying the trilinear vertex with the top-loop box that does not, giving the quadratic whose minimum locates the least sensitive value of \(\kappa_{\lambda}\), and the normalization that puts the rate three orders of magnitude below single production. The location of the minimum and the absolute rate are stated in this subsection on the authority of the experimental compilations and are not derived here; the two loop amplitudes belong in Appendix A with the single-production ones already recorded as owed. The published constraint on \(\kappa_{\lambda}\) quoted below is a measurement and needs no derivation.
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The perturbative correction to the Gross–Llewellyn Smith sum rule. The parton-model value of 3 is derived in the text above; what is still owed is the calculation of the coefficient function that multiplies it by \(1-\alpha_{s}/\pi-\dots\), order by order, and the demonstration that the same coefficient function is the one appearing in the Bjorken sum rule for polarized scattering. That is what makes a measurement of this integral a competitive determination of the strong coupling, and it belongs in Appendix A alongside the other sum rules.
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The nuclear modification of parton distributions, stated as the open problem it is: no derivation from quantum chromodynamics currently reproduces the observed dependence on the momentum fraction. The candidate mechanisms — nucleon swelling, pion-cloud enhancement, multi-quark clusters, and a dependence on the local nuclear density rather than on the mass number — each account for part of the shape and none for all of it, and they are not mutually exclusive. What is owed in Appendix A is a statement of what each predicts and of which measurements discriminate among them.
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The Ellis–Jaffe sum rule, the theoretical number the EMC measurement contradicts. What is owed is the derivation of the expectation for the first moment of the polarized proton structure function from the octet axial charges: the isovector combination fixed by the neutron beta-decay ratio, the octet combination fixed by the hyperon semileptonic decays under the flavour symmetry of the baryon octet, and the assumption of an unpolarized strange sea that sets the singlet piece equal to the octet one — together with the leading perturbative correction, which lowers the parton-model value. Only that chain shows why the expected number is near \(0.19\) and which of its assumptions the measurement actually falsifies. It belongs in Appendix A beside the other sum rules; this book's bibliography carries no entry for the original Ellis–Jaffe paper.
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The decomposition of the proton spin into quark spin, gluon spin and orbital contributions, and the reason the split is ambiguous. What is owed is the demonstration that the gauge-invariant decomposition of the angular momentum of a gauge theory into a “spin” and an “orbital” part is not unique — different decompositions differ by terms that vanish for a free field and do not vanish here — so that a measured \(\Delta G\) has meaning only relative to a stated convention and factorization scheme. It belongs in Appendix A with the operator definitions written out.
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The thermal-spike calculation of bubble nucleation: the temperature field around an ionising track from the heat equation with a line source, the fraction of the deposited energy thermalised within the critical radius before the bubble can grow, and the resulting threshold in \({[}\dd E/\dd x{]}\) as a function of the superheat. It belongs in Appendix A; the derivation above establishes the barrier and the mechanism but treats the energy delivery as instantaneous and perfectly local.
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Experiment: Brownian Motion and Avogadro's Number: all derivations of this chapter are pending.
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Molecular origin of the agitation: the unbalanced momentum delivered by molecular impacts on a grain of radius \(a\), and the reason the relative size of the resulting fluctuation grows as the grain shrinks
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The rotational displacement law for a sphere, in which the mean square angular displacement grows linearly in time with a coefficient carrying the inverse cube of the grain radius rather than its first power — the different power being what makes agreement between the two routes an independent check and not a repetition
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The velocity autocorrelation of the Ornstein–Uhlenbeck process obtained from the Langevin equation, its short-time ballistic limit, and the crossover to the diffusive law at the momentum relaxation time; and separately the hydrodynamic-memory kernel that replaces it for a sphere in a dense fluid
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Experiment: Bose–Einstein Condensation: all derivations of this chapter are pending.
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The polarization-gradient (Sisyphus) mechanism: the spatially modulated light shifts of the ground-state sublattice in a standing wave of varying polarization, the optical pumping that preferentially transfers an atom from a hill to the bottom of the next well, the resulting friction coefficient, and the recoil energy as the residual floor
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Interference of two overlapping expanding condensates: the fringe spacing set by the relative momentum of the two sources, and the emergence of a definite relative phase from repeated measurement on a state of fixed total atom number, in which no relative phase exists before the first detection
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The Bose–Hubbard description of atoms in a deep optical lattice, the tunnelling and on-site interaction parameters expressed through the lattice depth and the scattering length, and the mean-field estimate of the ratio at which the compressible superfluid gives way to the gapped, integer-filled insulator
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Quantum Optics and the Photon: all derivations of this chapter are pending.
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The shape of the pump–probe signal for two fragments separating on a repulsive potential energy surface: the classical clocking relation that maps probe detuning onto internuclear separation, the wave-packet computation of the fluorescence yield as a function of delay, and the extraction of the free-fragment appearance time from the saturating on-resonance curve
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The two-state model of alkali-halide dissociation: the diabatic covalent and ionic curves and the adiabatic well their crossing produces, the Landau–Zener probability of remaining on the adiabatic branch at each pass, and the resulting periodic recurrences with a geometrically decaying envelope and a step-wise product signal
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Experiment: The Quantum Hall Effect: all derivations of this chapter are pending.
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Laughlin's gauge argument: that adiabatically threading one flux quantum through a Hall cylinder is a gauge transformation and must map the spectrum to itself, so that its only effect is to transfer an integer number of electrons from one edge to the other; the resulting Hall conductance as that integer times the square of the elementary charge over Planck's constant, with no reference to the Hamiltonian; and the requirement of a mobility gap at the Fermi level that makes the process adiabatic — together with the topological restatement in which the integer is the Chern number of the occupied states, which cannot change under any smooth deformation of the sample
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Laughlin's correlated wavefunction for filling one over an odd integer, an incompressible liquid rather than a filled single-particle band; the plasma analogy that fixes its density and its excitation gap; the antisymmetry requirement that forces the exponent to be odd; and the hierarchy or composite-fermion construction that accounts for the remaining observed fractions
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The quasihole of the correlated ground state and the determination of its charge from the deficit of density produced by adiabatically threading one flux quantum, together with the plasma analogy that makes the deficit exact; and the Schottky formula for the noise of a Poisson stream of independent carriers, whose proportionality to charge times current is what converts a noise measurement into a charge measurement
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Landau quantization of a linear rather than quadratic dispersion, which gives level energies proportional to the square root of the product of level index and field and places one level exactly at zero energy; the shared occupation of that zero level by electrons and holes, which shifts the filling sequence by one half; the fourfold spin and valley degeneracy of the observed steps; and the Berry phase of the cyclotron orbit that appears as the offset in the Shubnikov–de Haas phase
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That a quantized Hall conductance requires only a gapped band structure with broken time-reversal symmetry and a non-zero topological invariant of the occupied bands, not a uniform magnetic field; the lattice model exhibiting this with zero net flux through the unit cell; and the mechanism by which magnetic dopants in a strong-spin–orbit thin film open the required gap in the surface states
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Experiment: Superconductivity: all derivations and data tables of this chapter are pending.
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The inference chain from the raw run: how a four-terminal measurement removes lead and contact resistance so that the reading belongs to the specimen; how the galvanometer sensitivity and the measuring current together convert into the quoted upper limit on the resistance; and the control observations that exclude the obvious alternative, a short circuit developing on cooling, by their dependence on measuring current and on applied magnetic field
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The inference from the measured isotope exponent: that a lattice frequency scales as the inverse square root of the ionic mass with the force constants, which are electronic, held fixed, so that an exponent of one half identifies the phonon energy as the prefactor of the pairing scale; and the corrections from the screened Coulomb repulsion, which enter the exponent and depress it below one half without altering the identification
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The coherence factors: that the matrix element for a process between quasiparticle states is a sum of two amplitudes whose relative sign depends on the behaviour of the perturbation under time reversal, so that the two combinations cancel and reinforce respectively; and the resulting opposite temperature dependences of ultrasonic attenuation and nuclear spin relaxation just below the transition, against the divergent density of states at the gap edge that both share
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What can and cannot be derived here: the demonstration that flux expulsion on field cooling requires a bulk, thermodynamic superconducting phase and excludes a percolating filamentary path; and the reading of the half-integer flux quantum trapped at a tricrystal ring as a sign change of the order parameter between directions. The mechanism itself is not derived anywhere in this treatise and is recorded as an open problem
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The one-phonon scattering cross section itself: the factor that weights each peak by the square of the projection of the polarization vector on the momentum transfer, by the Bose population of the mode, and by the Debye–Waller factor, so that the measured intensities constrain the eigenvectors of the dynamical matrix and not only its eigenvalues
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The mode sum for the mean square displacement, giving \(2W\) in terms of the measured dispersion relation, its evaluation in the Debye model with the zero-point term separated from the thermal one, and the resulting temperature dependence of the attenuation at fixed momentum transfer
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The probability distribution of the triplet phase sum: the random-atom model in which the atomic positions are independent and uniform over the cell, the central-limit argument that makes the joint distribution of three normalized structure factors Gaussian, and the conditional distribution of the phase sum that follows from it, giving the hyperbolic tangent in the centrosymmetric case and the von Mises form in general
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The contrast transfer function of a defocused electron microscope: the weak-phase-object approximation, the phase shift introduced by defocus and by spherical aberration, and the resulting oscillating transfer function whose zeros must be filled in by combining images taken at different defocus
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Experiment: Dark-Matter Searches: the standard halo model and the nuclear form factor that together convert a measured recoil spectrum into a bound on a cross-section, and the statistical construction of a 90 per cent confidence exclusion contour in the presence of an estimated background, are pending. The kinematic and rate arguments of this chapter are derived inline.
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The form factor and the recoil spectrum: derivation of the Helm form factor as the Fourier transform of a uniform nuclear density convolved with a surface term, its first zero in relation to the recoil energies of interest for xenon, and the resulting distortion of the differential rate away from the pure exponential that the maximum-recoil relation alone would give; together with the detector response function that converts a recoil energy into an expected number of scintillation photons and ionization electrons, which is what actually sets the threshold
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The quantitative incompatibility: the calculation that converts a modulation amplitude measured in sodium iodide into an allowed region in the mass–cross-section plane, including the quenching factors for sodium and for iodine recoils, which are measured quantities with their own uncertainties and which set the recoil-energy scale of the claimed region; and the comparison of that region against the exclusion curves of the xenon experiments, together with an enumeration of the non-standard halo and interaction assumptions — streams, inelastic scattering, isospin-violating couplings — that have been proposed to reconcile them and the reasons each is now disfavoured
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The thermal relic cross-section: solution of the Boltzmann equation for a species annihilating in an expanding universe, freeze-out when the annihilation rate falls below the expansion rate, and the resulting inverse proportionality of the relic abundance to the annihilation cross-section, which is what fixes the benchmark value quoted above from the abundance measured in the dark sector chapter; and the reason the required value falls near a weak-scale cross-section, which is the coincidence that motivated a generation of searches
Missing write-ups
Registered experiments that still lack one or more
required headings.