Thinking in Waves Episode 62 / Part XX — The Things That Tunnel
Fusion and superconductivity turned out to be the same integral
The things that tunnel ──
counting everything in decades \(=G/\ln 10\)
\(G=\frac{2}{\hbar}\int\sqrt{2m(V-E)}\,dr\) ── this one line computes fusion, cold fusion, Josephson junctions, STM and \(\alpha\) decay. Only the shape of \(V(r)\) changes.
★★ As this series always does, we count decades ── STM costs 5, a reactor 3, room-temperature \(\mathrm{D_2}\) 83.
★★★ And the “waves add coherently” rescue of cold fusion can be refuted by counting alone ── all the deuterium in Earth’s oceans still falls 9 orders short.
“Superconductivity and fusion are all just tunnelling, aren’t they?” ── Yes, exactly.
★ Part XIX found that the limits of a measurement are set by where you cut. This time it is not the cut but the barrier.
★★ And for the first time in this series, the page contains calculators you can drive yourself.
01One integral, three famous formulas
| barrier | \(V\) | \(G\) | the knob | where it shows up |
|---|---|---|---|---|
| Coulomb | \(kZ_1Z_2/r\) | \(\sqrt{E_G/E}\) | \(E^{-1/2}\) | fusion, \(\alpha\) decay |
| square | \(V_0\) (width \(d\)) | \(\frac{2d}{\hbar}\sqrt{2m(V_0-E)}\) | linear in \(d\) | STM, Josephson |
| triangular | \(\phi-eFx\) | \(\frac{4\sqrt{2m}}{3\hbar eF}\phi^{3/2}\) | \(F^{-1}\) | field emission |
★ Numerical engine against analytic solutions: Coulomb agrees to \(10^{-7}\), square and triangular to \(10^{-9}\). The Gamow factor, the square barrier and Fowler–Nordheim are three faces of one integral.
★★ And the Gamow energy \(E_G=2\mu c^2(\pi\alpha Z_1Z_2)^2\) depends only on mass and charge ── no nuclear physics enters at all.
| reaction | \(\mu c^2\) [MeV] | \(E_G\) [keV] | our \(B_G\) | literature \(B_G\) |
|---|---|---|---|---|
| p + p | 469.14 | 493.1 | 22.2065 | 22.20 |
| d + d | 937.81 | 985.8 | 31.3969 | 31.3970 |
| d + t | 1124.65 | 1182.2 | 34.3826 | 34.3827 |
| d + ³He | 1124.56 | 4728.3 | 68.7626 | 68.7508 |
★★★ Agreement to four figures (literature: Bosch–Hale 1992). The entire “difficulty” of fusion sits in this one number.
02Thermonuclear fusion ── try moving it
Thermonuclear calculator
The core of §02
★ It is not the average particle that reacts. The product of the Maxwell tail (\(e^{-E/kT}\)) and the sharply rising tunnelling probability (\(e^{-\sqrt{E_G/E}}\)) makes a Gamow peak, and only that burns.
★★ Even at the Sun’s centre (1.35 keV = 15.7 million K) the peak is at 6.1 keV ── only the tail, 4.5× the average.
★★★ The reactivity matches the literature to within 1 %: \(\langle\sigma v\rangle_{dt}=1.089\times10^{-22}\) at 10 keV (lit. \(1.1\times10^{-22}\)) and \(4.244\times10^{-22}\) at 20 keV (lit. \(4.3\times10^{-22}\)) m³/s; the peak is at 66.5 keV (lit. ≈64).
03The same Coulomb barrier also gives \(\alpha\) decay
Alpha decay calculator
The core of §03 ── the \(E^{-1/2}\) in the exponent decides everything
★ ²³⁸U (4.27 MeV) and ²¹²Po (8.95 MeV) differ only by a factor 2.1 in energy.
★★★ Their lifetimes differ by 24 orders of magnitude ── 4.47 billion years versus 0.3 microseconds.
★★ Our calculation gives \(\log_{10}(T/\mathrm{s})=17.16\) for ²³⁸U against a measured 17.15. That a crude attempt frequency and a crude nuclear radius still land on it is because the exponent dominates everything ── the most extreme case of Episode 53’s “quantities that only move logarithmically.”
04Tunnelling in solids ── STM and superconducting junctions
Square-barrier calculator
The core of §04 ── why single atoms are visible
★ At \(\phi=4.5\) eV (a metal work function), moving the tip 1 Å closer multiplies the current by 8.8 ── exactly 0.94 decades.
★★★ The STM rule of thumb “one decade per ångström” is this formula itself, and it is why a single atom’s corrugation is visible.
★★ A Josephson junction (AlOx, \(\phi\approx2\) eV) costs 6.3 decades at 1 nm; another 0.5 nm of oxide costs 3 more ── which is why superconducting qubit fabrication demands ångström control.
★ The Coulomb barrier goes as \(E^{-1/2}\), but a square barrier is linear in \(d\) ── so with STM, decades can be bought linearly in distance. In Episode 54’s terms, it is the most generous counterpart there is.
05★★★ Cold fusion ── counting how many decades are missing
The barrier-lowering effect of electrons is captured experimentally by a single screening energy \(U_e\) (\(\sigma_{\text{scr}}(E)=\sigma_{\text{bare}}(E+U_e)\)). Inside a bound molecule \(E\approx0\), so effectively \(E\to U_e\).
★★ Calibrating this model on a single point (\(U_e=36\) eV reproduces Koonin–Nauenberg’s \(3\times10^{-64}\,\mathrm{s^{-1}}\) for \(\mathrm{D_2}\)) automatically yields \(10^{9.7}\,\mathrm{s^{-1}}\) for the muonic molecule (literature ≈\(10^9\)) ── one calibration point spanning 73 orders of magnitude to within a factor of a few. So the model is usable.
Cold fusion calculator ── try moving \(U_e\)
| system | \(U_e\) [eV] | source | \(G\) | \(\lambda\) [/s/pair] | per mole [/s] |
|---|---|---|---|---|---|
| \(\mathrm{D_2}\) gas target (meas.) | 25 | Raiola 2002 | 198.6 | \(10^{-78.3}\) | \(10^{-54.5}\) |
| adiabatic limit (theory) | 27 | textbook | 191.1 | \(10^{-75.0}\) | \(10^{-51.3}\) |
| deuterated Ti | 36 | Raiola 2002 | 165.5 | \(10^{-63.9}\) | \(10^{-40.1}\) |
| deuterated Pd | 250 | Raiola 2002 | 62.8 | \(10^{-19.3}\) | \(10^{+4.5}\) |
| deuterated Ta | 309 | Kasagi 2002 | 56.5 | \(10^{-16.6}\) | \(10^{+7.2}\) |
| Au/Pd/PdO multilayer | 601 | Raiola 2002 | 40.5 | \(10^{-9.6}\) | \(10^{+14.1}\) |
The core of §05 ── this is the genuinely interesting part
★ Experimental upper limits are \(10^{-24}\) to \(10^{-23}\,\mathrm{s^{-1}}\) per pair (Jones 1989 and others). The \(U_e\) corresponding to that is 194 eV.
★★★ Yet the measured \(U_e\) for deuterated Pd is 250 eV ── above the limit.
★★ Taken at face value, the model therefore predicts that cold fusion should be observed. It is not.
★★★ So the conclusion is not “cold fusion does not exist” but “extrapolating \(U_e\) down to \(E\to0\) breaks the approximation” ── in Episode 55’s language, a conditional value being used outside its condition.
| what is broken | detail |
|---|---|
| ① extrapolation | \(U_e\) is measured in beam experiments at \(E=5\)–20 keV. The shift \(E\to E+U_e\) is valid only for \(U_e\ll E\); \(E\to0\) is outside any guarantee |
| ② electron density | producing 250 eV by static screening needs a screening length of about 0.06 Å; metallic electron densities fall orders short |
| ③ distance | the d–d distance in a metal is about 2.9 Å, far larger than \(\mathrm{D_2}\)’s 0.74 Å ── the barrier is not lowered but wider |
06★★★ “Does the wave picture give anything?” ── three honest counts
First the premise. Treating fusion as a wave is exactly what Gamow did in 1928, and that in itself is not new ── tunnelling is a wave leaking where classical particles cannot go. With that said, here are three things the wave language turns into something countable.
(A) d+t is chosen not for its barrier but for a resonance
| reaction | \(B_G\) | \(E_{cm}\) | Gamow factor | \(\sigma\) [mb] | \(S(E)\) [keV·b] |
|---|---|---|---|---|---|
| d + t | 34.38 | 50 keV | 7.73e−3 | 3970 | 25675 |
| d + d | 31.40 | 50 keV | 1.18e−2 | 33.2 | 141 |
★ \(B_G\) is 2.99 larger for d+t ── tunnelling is 1.53× worse for it.
★★★ And yet the cross-section is 120× larger. The whole difference lives in \(S(E)\) (182×).
★★ d+t’s advantage is the ⁵He resonance at 64 keV ── a standing wave state on the nuclear side. “Reactors use d+t because its barrier is lower” is, strictly, backwards.
(B) Is the muon “just barely” heavy enough?
Replace the electron with a particle of mass ratio \(x\): the molecule shrinks to \(1/x\), \(U_e\) grows by \(x\), and \(G\) shrinks by \(1/\sqrt{x}\).
| mass ratio \(x\) | molecule size | \(G\) | \(\lambda\) [/s] | |
|---|---|---|---|---|
| 1 | 0.741 Å | 164.6 | \(10^{-63.5}\) | electron (\(\mathrm{D_2}\)) |
| 30 | 0.0247 Å | 30.0 | \(10^{-0.7}\) | too slow |
| 85 | 0.0087 Å | 17.9 | \(10^{6.0}\) | ★ threshold |
| 195.7 | 0.0038 Å | 11.8 | \(10^{9.7}\) | muon (\(\tau=2.2\,\mu\)s) |
| 273 | 0.0027 Å | 10.0 | \(10^{10.9}\) | pion (\(\tau=26\) ns) |
★★★ The threshold for useful catalysis is a mass ratio of about 85. The muon sits at 195.7, a factor 2.3 above ── not marginal.
★★ But there is no charged particle between the electron and the muon. The only charged particles above the threshold are the muon and the pion, and the pion decays in 26 ns.
★ So muon-catalysed fusion has exactly one candidate ── the muon is the only particle in nature satisfying both mass and lifetime. And what sets that threshold is the \(x^{-1/2}\) law of leakage.
(C) ★★★ If it is a wave, it should add coherently ── does that close the gap?
This is the strongest thing this episode can say. Claims of the form “cold fusion is explained by a collective wave effect” can be judged without looking inside the model at all, by counting.
| value | |
|---|---|
| theory (\(U_e=27\) eV, \(\mathrm{D_2}\) geometry) | \(10^{-75.0}\) /s/pair |
| experimental limit (Jones 1989 etc.) | \(10^{-23.0}\) /s/pair |
| gap to close | 52 orders |
| coherent \(N\) required (even with \(N^2\) superradiance) | \(10^{52}\) |
| one mole | \(10^{23.8}\) ── 28 orders short |
| all deuterium in Earth’s oceans | \(\approx10^{43}\) ── 9 orders short |
The core of §06
★★★ Applying the largest conceivable coherence — perfect superradiance — to every deuteron in Earth’s oceans still falls 9 orders short.
★★ So the “waves add coherently” rescue fails for any specific model whatsoever ── it can be refuted by counting.
★ But this is not a proof that cold fusion does not exist. It is the narrower claim that a coherence mechanism cannot explain it, and the only road left is ① from §05.
| there are only three knobs | required | what exists | verdict |
|---|---|---|---|
| ① raise the screening \(U_e\) | 194 eV | theory 27 eV / metals measured 250 eV | ? |
| ② shrink the distance \(R\) | 0.103 Å | \(\mathrm{D_2}\) 0.741 Å / PdD ≈2.9 Å | × |
| ③ raise the energy \(E\) | 194 eV = 2.25 million K | room temperature 0.025 eV | × |
★★★ Only ① remains, which is why every cold-fusion argument converges on “how far do you trust \(U_e\)”.
★★ And ① is measurable: lower the energy and watch whether the apparent \(U_e\) stays constant. If it does, the model holds; if it drifts, the extrapolation is broken ── exactly the form of question Episode 55’s handbook demands.
07All of it on one page ── decades \(=G/\ln 10\)
| phenomenon | barrier | \(G\) | decades of suppression | the knob |
|---|---|---|---|---|
| d+t (20 keV, reactor) | Coulomb | 7.7 | 3.3 | temperature |
| STM (0.5 nm vacuum) | square 4.5 eV | 10.9 | 4.7 | distance |
| dd-μ molecule | Coulomb + screening | 11.8 | 5.1 | the muon’s mass |
| Josephson (1 nm) | square 2 eV | 14.5 | 6.3 | oxide thickness |
| field emission (3 GV/m) | triangular 4.5 eV | 21.7 | 9.4 | electric field |
| d+d (Sun, 1.35 keV) | Coulomb | 27.0 | 11.7 | temperature |
| ²³⁸U \(\alpha\) decay | Coulomb | 88.0 | 38.2 | ── nothing to turn |
| \(\mathrm{D_2}\) (room temperature) | Coulomb + screening | 191 | 83 | ── no knob |
The core of §07 ── this table is the whole thing
★★★ Top to bottom, one and the same integral. Only \(V(r)\) differs.
★★ The “usable” phenomena cost around 10 decades; the “unusable” ones cost 80. Whether there is a knob in between is the whole difference.
★ The muon really does take 83 decades down to 5. That is why it works. (It works, but with a 2.2 μs lifetime it catalyses only ~150 times, which misses breakeven ── that is a separate problem, not a tunnelling one.)
Exercises
- §04 says STM buys decades linearly in distance. What is that in the language of Episodes 53 and 54?
Show answer
The most generous way of buying range there is.
★ Episode 53: “range can only be bought logarithmically” (ten times the observation \(T\) buys one decade). ★★ But a square barrier has \(G\propto d\), so shrinking the distance linearly buys decades linearly ── one decade per ångström.
★★★ The Coulomb barrier has \(G\propto E^{-1/2}\), so doubling the decades needs four times the energy ── that side belongs to Episode 53’s “only as a power”.
★★ So “what you must buy to gain decades” is decided by the barrier’s shape ── STM reaching atomic resolution while fusion demands 100 million K is only a difference of exponent inside the same formula. - §06 (C) says the claim can be refuted by counting. Why is that a strong argument?
Show answer
Because it does not depend on the mechanism at all.
★ The usual rebuttal knocks down one proposed mechanism at a time, and has to start over whenever a new one appears.
★★ But coherent amplification is bounded by \(N\) for any mechanism (even superradiance gives \(N\)). So counting the required \(N\) against the available \(N\) ends it.
★★★ This is the strongest form of Episode 55’s “count the assumptions” ── judging by the ceiling of what an assumption could deliver, without examining the assumption.
★ Its limits are equally sharp: it says nothing about non-coherence mechanisms (a genuinely different screening, say). ★★ Unless you write down what it does and does not refute, this argument gets abused. - §05 finds the measured \(U_e\) exceeding the limit. Is that a contradiction?
Show answer
Not a contradiction ── a broken extrapolation.
★ \(U_e\) is obtained by measuring how fast \(\sigma\) rises at \(E=5\)–20 keV. ★★ There \(U_e/E\sim0.02\), and the shift \(E\to E+U_e\) is a correct first-order approximation.
★★★ But at \(E\to0\), \(U_e/E\to\infty\) ── we are five decades outside where that first-order form applies.
★ In Episode 57’s language: this is an edge. The leading term of an expansion was carried outside the range where the expansion works.
★★ In Episode 60’s language: a value was used without saying where it was cut. \(U_e\) is “the \(U_e\) measured at 5–20 keV”, not a constant.
Summary — one integral, counting 83 decades
“Superconductivity and fusion are all just tunnelling, aren’t they?” ── Yes.
★★★ \(G=\dfrac{2}{\hbar}\displaystyle\int\sqrt{2m(V-E)}\,dr\), \(\;\)decades \(=G/\ln 10\).
Only \(V(r)\) changes.
★ All 18 checks pass: \(B_G\) agrees with the literature to four figures, \(\langle\sigma v\rangle_{dt}\) to within 1 %, ²³⁸U’s \(\alpha\) decay gives \(\log_{10}(T/\mathrm{s})=17.16\) against a measured 17.15, STM gives 0.94 decades per ångström, and dd-μ gives \(5.2\times10^9\) against a literature \(1.5\times10^9\).
★★ Cold fusion reduces to \(U_e\) alone. In a model calibrated on one \(\mathrm{D_2}\) point that reproduces the muonic molecule 73 orders away, the \(U_e\) matching the experimental limit is 194 eV ── while deuterated Pd measures 250 eV. ★★★ So the conclusion is not “cold fusion does not exist” but “extrapolating \(U_e\) to \(E\to0\) is broken” ── a question that can be measured.
★★★ And three things the wave picture yielded. (A) d+t’s advantage is not the barrier but the ⁵He resonance (its \(B_G\) is in fact 1.53× worse; the difference is entirely \(S(E)\)). (B) The catalysis threshold is a mass ratio of 85, and the only charged particles above it are the muon and the pion, the latter dying in 26 ns ── hence exactly one candidate. (C) A coherence mechanism falls 9 orders short even with every deuteron in Earth’s oceans in perfect superradiance.
★ (C) is the strongest statement here ── an entire class of explanations refuted by counting, without inspecting a single mechanism. ★★ But what was refuted is “coherence mechanisms”, not “cold fusion”. Without writing that distinction down, the argument gets abused (Exercise 2).
★ And §06 (B) showed that it is the one option nature provides.
★★ Next we dig into that “only one” ── what is a catalyst, and what does it mean to help something tunnel? ★ Episode 45’s Bogoliubov coefficient, Episode 58’s window sidelobes, and this episode’s muon ── we ask whether “reaching in from outside to change an exponent” is the same operation in all three.
This document is Episode 62 of the “Thinking in Waves” series, opening Part XX. The computations are in kenshou/tunnel.js (zero dependencies, no randomness, 18 checks). This is the first episode in the series to carry calculators the reader can drive (the sliders in §§02–05); every displayed value is computed on the spot from the same formulas as that script. Not discoveries of this episode: the WKB approximation and the Gamow factor (Gamow 1928), the Gamow peak, Fowler–Nordheim, Geiger–Nuttall, the screening-energy \(U_e\) framework, muon-catalysed fusion, Koonin–Nauenberg’s \(\mathrm{D_2}\) rate of \(3\times10^{-64}\,\mathrm{s^{-1}}\), and the ⁵He resonance in d+t ── all standard, known results. What this episode did was re-express them on the single yardstick \(G/\ln10\) (decades of suppression), ① verify that one \(\mathrm{D_2}\) calibration point reproduces the muonic molecule 73 orders away, ② count the \(U_e\) matching the experimental limit as 194 eV, and ③ count the \(N\) a coherence mechanism would need as \(10^{52}\), leaving all of Earth’s deuterium 9 orders short. Limits of the model: the rates in §§05–06 use \(\lambda=A_{dd}|\psi(0)|^2\) with \(|\psi(0)|^2\approx e^{-G}/V\), a crude estimate with a factor-of-a-few uncertainty in absolute value (calibrated on one \(\mathrm{D_2}\) point). The decade-counting is unaffected, but boundary values such as “194 eV” are model-dependent. The threshold of 85 in §06 (B) uses the convenience criterion \(\lambda>10^6\,\mathrm{s^{-1}}\) and moves if that changes. The \(\alpha\) decay in §03 uses an attempt frequency \(v/2r_n\) and a radius \(1.2(A^{1/3}+4^{1/3})\) fm, both crude. This episode too uses only literature values and analytic formulas; it has not been applied to raw measured data. Main references: Bosch & Hale, Nucl. Fusion 32, 611 (1992); NRL Plasma Formulary; Kasagi et al., Eur. Phys. J. A (2002); Raiola et al. (2002); Koonin & Nauenberg, Nature 339, 690 (1989); Jones et al., Nature 338, 737 (1989); CODATA 2018. ── To print, use your browser’s Print → Save as PDF.