Rutherford-Ernest: Primer

Source note (carried from Steps 1 and 3). No Source gap for the main works: all seven documents have text, and the deep readers read the equations from page images. The project holds Rutherford's two Harper's articles (January 1904 and February 1905), Radio-Activity (1904), Radioactive Transformations (1906), the 1911 scattering paper (Phil. Mag. 21, 669; a 2002 web transcription, not a journal scan), Radioactive Substances and Their Radiations (1913) and Radiations from Radioactive Substances (1930, with Chadwick and Ellis). After the quality check (QE, 2026-10-09) the 1930 book's chapters on artificial disintegration and on attempts to change decay rates (Ch. IX-XI), its beta-energy section (§88-94), haloes and ordinary-matter pages, the 1913 book's alpha-ray chapter (re-run after the first read was lost), its heat and temperature sections, and the 1904 book's Ch. X (sec. 199-207) were all deep-read, with page images where the equations needed them. Limits that still travel with every conclusion below: the remaining chapters of the three books (measurement methods and spectra) are read only as page-by-page digests, and the 1919 Rutherford-Compton gravity letter was seen only as an abstract [Arbiter, QE revision, Source gap update]. Tags: [read] project documents and deep reads; [web] with URL; [memory] with confidence; [calc] arithmetic done for this primer (all of it is in example.py).

Framing. This is not an alternative-science project. It is the founding record of nuclear physics, written by the man who did most of it. Six of the seven central claims are now textbook physics in their core [Arbiter]. It belongs in this review for three reasons: it is the standard against which other projects (Moray, Le Bon, Becquerel, Morray-TH) are measured; it shows how a first-rate scientist reasoned from measurement, made numerical errors and corrected them; and it contains a short list of anomalies and untested questions, and a "detonator" passage about externally triggered decay that readers of the Moray and Le Bon projects will meet again. Interested parties: Rutherford wrote every document and originated both theories under test, and most of the confirming experiments came from his own students (Soddy, Barnes, Geiger, Marsden, Gates, Bronson, Royds, Chadwick); independent confirmation is named where it exists.

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1. The idea in one paragraph

Around 1900 a few heavy elements (uranium, thorium, radium) were found to give off invisible rays and a steady trickle of heat, forever as far as anyone could tell, with no fuel. Rutherford and Frederick Soddy explained this in 1902-03: the atoms of these elements are not permanent. Each atom has a fixed chance per second of breaking up into an atom of a different element, throwing out a fast particle as it does, and the chain of break-ups (uranium to radium to radon to lead, with helium shed along the way) is what the instruments see. The heat comes from inside the atom and is about a million times larger, per atom, than any chemical reaction. In 1911 Rutherford used the fast "alpha" particles from radium as probes: a few of them bounced almost straight back from a thin gold foil, which could only happen if nearly all of an atom's positive charge and mass sat in a tiny central knot, now called the nucleus. From these two ideas came the age of the earth from radioactive heat and helium, the first man-made transmutation (nitrogen hit by alpha particles, 1919), and the modern picture of the atom. Rutherford also left some questions open: whether some outside agency might "detonate" the break-up while the energy stays the atom's own, and whether gravity could change the rate. Triggering by radiation was tried many times between 1903 and 1928 and never found; gravity is the part of the record still marked untested.

2. What you need to know first

Exponential decay and half-life. If every atom has the same fixed probability lambda (the "decay constant", units 1/s) of breaking up in each second, independent of its age and its neighbours, then a sample of N atoms loses lambda N atoms per second: dN/dt = -lambda N. The solution is N(t) = N0 e^(-lambda t), where N0 is the number at t = 0. The time for half to go, the half-life, is T_half = ln 2 / lambda; the mean life is 1/lambda. The decay rate lambda N is called the activity. Rutherford's "half value period" is the half-life. Read: Krane, Introductory Nuclear Physics (Wiley, 1988), the chapter on radioactive decay; or the radioactivity chapter of any modern-physics text (Serway, Moses and Moyer, Modern Physics).

Chains of decays (parent and daughter). If A decays into B and B decays into C, then dN_B/dt = lambda_A N_A - lambda_B N_B: B is made by A and lost by its own decay. Solving these linear ODEs (the "Bateman equations", Rutherford's 1913 book prints them) gives curves that rise, peak and fall. When the parent lives much longer than the daughter, the daughter reaches "secular equilibrium": lambda_A N_A = lambda_B N_B, equal activities. That is how Rutherford and Boltwood predicted that old uranium ores hold radium in a fixed ratio, 3.4 x 10^-7 by weight [read: claims.md; Arbiter R5]. Read: Krane, same chapter.

Coulomb's law in Gaussian units. The 1911 paper uses centimetre-gram-second "electrostatic" units, where the force between charges q1 and q2 a distance r apart is F = q1 q2 / r^2 (no 4 pi epsilon_0), the potential energy is U = q1 q2 / r, and charge is in e.s.u. (statcoulombs). The electron charge is e = 4.803 x 10^-10 e.s.u. [memory, high]; Rutherford used 4.65 x 10^-10 [read: deep/structureatom section 0]. A handy shortcut: e^2 = 1.44 MeV x fm, where 1 MeV = 1.602 x 10^-6 erg and 1 fm (femtometre) = 10^-13 cm. So the potential energy of charges Z1 e and Z2 e at r femtometres is 1.44 Z1 Z2 / r MeV [memory, high]. Read: Griffiths, Introduction to Electrodynamics, the appendix on units.

Central-force orbits and the hyperbola. A particle moving under a force that points along the line to a fixed centre conserves angular momentum about that centre (m v p = constant, where p is the "impact parameter", the distance by which the incoming straight-line path would miss the centre). With an inverse-square repulsion the path is a hyperbola, with the centre at the outer focus. Energy and angular-momentum conservation together give the deflection angle. Read: Taylor, Classical Mechanics (University Science Books, 2005), the chapters on central forces and on collision theory; or Thornton and Marion, Classical Dynamics, the chapter on central-force motion.

Cross-section and the "fraction scattered". A thin foil of thickness t holding n atoms per cm^3 presents n t atoms per cm^2 to a beam. If each atom deflects any particle that passes within distance p of it, the fraction deflected is (pi p^2) x (n t): the area each atom "blocks" times how many atoms per unit area. The area pi p^2 is a "cross-section", sigma. This is the whole of the probability argument in the 1911 paper. Read: Taylor, the collision-theory chapter.

Random walks. If a particle receives many small, random sideways kicks of typical size theta, its total angle after m kicks is about theta x sqrt(m), and the chance of a total angle larger than phi falls like a Gaussian, e^(-phi^2/theta_total^2). Gaussian tails fall off far faster than any power law. That one fact decides between Thomson's atom and Rutherford's (Section 5, Step 5). Read: any probability text on the central limit theorem; Reif, Fundamentals of Statistical and Thermal Physics, the opening chapter on random walks.

Mass, energy and calories. In a decay the products weigh slightly less than the parent; the missing mass Delta m appears as energy Q = Delta m c^2. Per atom this is a few MeV for nuclear decay against a few eV for chemistry. The heat unit of 1900 is the gram-calorie: 1 cal = 4.184 J; "100 cal per hour" = 0.116 W. Read: Krane, the chapter on nuclear masses and alpha decay.

3. The textbook view

What the rays are. Alpha particles are helium-4 nuclei (charge +2e, mass 6.645 x 10^-24 g), emitted with a single sharp energy for each emitter (4-9 MeV, speeds 1.4-2.1 x 10^9 cm/s, about 5-7 percent of the speed of light c). Beta particles are electrons, emitted with a spread of energies. Gamma rays are high-energy photons [memory, high]. Rutherford and Royds proved alpha = helium in 1908-09 by letting alphas pass through a thin glass wall (under 1/100 mm) into an empty tube and watching the helium spectrum appear: the yellow helium line after two days, the whole spectrum after six; helium put in place of the emanation, as a control, did not leak through. The 1913 book: "There can thus be no doubt that the alpha particle becomes a helium atom when its charge is neutralised" [read: deep/Radioactive-Substances, PDF pp.131-192, via the Arbiter QE revision section 9; web: https://web.lemoyne.edu/Giunta/royds.html]. The same chapter gives the alpha's charge measured directly by Rutherford and Geiger (1908) as 9.3 x 10^-10 e.s.u., twice the unit charge, and 3.4 x 10^10 alphas per second from one gram of radium (modern 3.7 x 10^10) [read; memory, high for the modern value].

The decay law. dN/dt = -lambda N, with lambda fixed for each nuclide. Rutherford's own worked case: thorium X, half-life about 4 days, so lambda = ln 2 / (96 h) = 0.00722 per hour; after one hour 1 - e^(-0.00722) = 1/139 of the activity has regrown in the stripped thorium (the book says 1/140); for the radium B to C chain the daughter peaks at t_max = ln(lambda_2/lambda_1)/(lambda_2 - lambda_1) = 35 minutes [read: Arbiter R5; physicist and mathematician computed the same].

Where the heat comes from. Each alpha decay releases Q = Delta m c^2. Q is shared between the alpha and the recoiling daughter nucleus in the inverse ratio of their masses, so the heat per decay is Q = E_alpha x A/(A - 4), where A is the parent's mass number. One gram of radium-226 in equilibrium with its short-lived products down to radium C' (polonium-214) has four alpha emitters, each decaying 3.7 x 10^10 times per second (this number defined the old unit, the curie). The four decay energies sum to 24.4 MeV, so the power is 24.4 x 3.7 x 10^10 x 1.602 x 10^-13 J = 0.145 W = 124 cal/h [calc, example.py Part B; alpha energies [memory, high]]. Rutherford and his contemporaries measured about 100 to 134 cal/h [Arbiter R1]. The total energy store in a gram of radium, decaying all the way to lead, is the mass difference: 35.3 MeV per atom = 3.6 x 10^9 cal/g, against Rutherford's 3.7 x 10^9 [Arbiter, item 5; calc]. Where his number comes from: the 1913 book derives 2.44 x 10^9 cal/g step by step for the emanation and its products (109 cal/h per curie, divided by the emanation's decay constant), then states "about 3.7 x 10^9 calories" for radium through radium F without the steps; scaling by the five alpha energies against three reproduces it (3.6 x 10^9) [read: 1913 book p.607; Arbiter QE revision section 2]. The 1904 first edition had given a range, 2 x 10^9 to 5 x 10^10 cal/g, "the minimum estimate is probably nearer the truth" [read: 1904 book sec. 205].

Beta decay does not give one energy. Alpha emitters release a single sharp energy; beta emitters release electrons with a continuous spread up to a maximum. Calorimetry settled what the heat is. Ellis and Wooster (1927) absorbed all the betas from radium E in a lead calorimeter and measured 350,000 +/- 40,000 electron-volts per disintegration: the mean of the spread (390,000 +/- 60,000), not the maximum (1,050,000). Meitner and Orthmann in Berlin, a different group, got 337,000 +/- 20,000 (1930). The 1930 book concludes that "the nucleus can break up with emission of an amount of energy that varies within wide limits. No satisfactory interpretation has yet been given" [read: 1930 book PDF pp.440-441, via the Arbiter QE revision section 3]. Ellis was a co-author of that book. Mainstream physics kept energy conservation by postulating an unseen particle, the neutrino (Pauli, 1930), first detected by Cowan and Reines in 1956 [web: https://en.wikipedia.org/wiki/Neutrino, via the arbiter]; Bohr had been willing to give up strict conservation [memory, high].

Rate invariance. For alpha and beta decay, lambda does not change measurably with temperature, pressure, chemical form, or electric and magnetic fields. The known exceptions are small and specific: electron-capture decays (where the nucleus swallows one of its own atom's electrons) change by 0.1 to 1 percent with chemical environment or pressure, and fully stripped ions can decay very differently (bare rhenium-187: half-life 32.9 years against 42 billion years for the neutral atom) [web: https://www.osti.gov/biblio/404092, Bosch et al., Phys. Rev. Lett. 77, 5190 (1996), via the arbiter; Physics World summary via the physicist].

Rutherford scattering. An alpha particle (charge 2e, mass m, speed u, kinetic energy K = (1/2) m u^2) passes a nucleus of charge Z e. Define:

• b = the head-on distance of closest approach, where all the kinetic energy has turned into potential energy: K = 2 Z e^2 / b, so b = 2 Z e^2 x 2 / (m u^2) = 2 Z (1.44 MeV fm)/K. Rutherford writes it b = 2 N e E/(m u^2) with N the number of unit charges at the centre and E the alpha's charge [read: deep/structureatom section 2].
• p = impact parameter; phi = deflection angle.

Then:

1. cot(phi/2) = 2p/b. (Rutherford's equation (1).)
2. Fraction of particles deflected by more than phi in a foil with n atoms/cm^3 and thickness t: f(>phi) = (pi/4) n t b^2 cot^2(phi/2). (Equation (3): just pi p^2 n t with p from (1).)
3. Number per unit area on a screen at distance r, for Q incident particles: y = Q n t b^2 / (16 r^2 sin^4(phi/2)). (Equation (5).) In modern language the differential cross-section is d sigma/d Omega = (b/4)^2 cosec^4(phi/2), where Omega is solid angle and cosec (the cosecant, also written csc) is cosec x = 1/sin x, so cosec^4(phi/2) = 1/sin^4(phi/2).
4. Closest distance on a path deflected by phi: d = (b/2)(1 + cosec(phi/2)).

All five of Rutherford's equations were re-derived by the deep reader with SymPy and found correct, and the quantum-mechanical Coulomb cross-section is identical to the classical one [read: deep/structureatom checks; memory, high]. Modern nuclear radii follow R = 1.2 fm x A^(1/3) [memory, high]: about 7 fm for gold, 10,000 times smaller than the atom (about 10^-8 cm).

The earth and the sun. The earth's surface heat flow is about 47 TW; geoneutrino measurements put about 20 TW of it down to radioactive decay of uranium, thorium and potassium, concentrated in the crust and mantle [web: Gando et al., Nature Geoscience 4, 647 (2011), via the physicist and arbiter]. The sun is powered by hydrogen fusion (Bethe 1938-39), which releases 0.7 percent of the rest-mass energy, about 1.5 x 10^11 cal/g [memory, high; calc, Arbiter R3].

4. Where Rutherford departs (and where his contemporaries did)

Rutherford is, for the most part, the textbook. The departures are particular numbers and inferences that were reasonable at the time and later changed. The table sets each beside the modern view.

Le Bon's quotation of Rutherford. Gustave Le Bon, who claimed priority for "universal radioactivity" from 1897 and is an interested party, quoted the French version of the St Louis address as saying Rutherford was in doubt whether radium's energy is "borrowed from the atom itself or from external sources". The sentence is real (French text p.55); the English text reads that a chemical theory is required "whether the view is taken that the energy is derived from the atom itself or from external sources". In both versions the same address goes on to argue against the external source [read: sites.md; research/2026-10-09-archive-org.md]. Anyone quoting this passage should quote all three together: the "doubt" sentence, the argument against external sources, and the detonator passage [Arbiter, Claim 7a]. Moray's booklets copy Le Bon's text (see the Morray-TH project's follow-ups).

Transmutation by ordinary means: claims made by others, and how they were answered. The 1930 book (Sec. 74) lists six 1920s claims that elements were changed by electric discharges or chemistry, without fast alpha particles. Each is given here in the claimant's terms, then the answer the book reports; the book's authors are not neutral, since their theory says such energies are far too small (chemical energies are a few electron-volts, nuclear binding "several million volts") [read: 1930 book PDF pp.346-349, via the Arbiter QE revision section 6].

An earlier episode (1913 book Sec. 116): Cameron and Ramsay (1907-08) reported neon and argon from radium emanation acting on water, and lithium from copper solutions. Rutherford and Royds repeated it and found only helium, tracing the neon to an air leak; Curie and Gleditsch found lithium in glass and quartz vessels but none in platinum ones [read: deep/Radioactive-Substances Part B]. Ramsay's own replies are not in the pages read; he was a Nobel chemist, not a fringe figure. Four of the six 1920s claims were answered by repeating the experiment or by the claimants' own withdrawal, not by dismissal. Paneth-Peters is the best-known 1920s antecedent of the 1989 palladium cold-fusion claims [memory, high], and its control (helium absorbed by glass from the air) is one any later helium claim has to meet.

5. Worked example: the gold-foil numbers

Question. Starting from Rutherford's own 1911 inputs, how often should an alpha particle be turned through more than 90 degrees by a thin gold foil, and does that match the "about 1 in 20,000" that Geiger and Marsden saw? Why can't many small deflections (Thomson's picture) do the same job? And what did the later scattering experiments actually establish about the size of the nucleus?

Inputs [read: deep/structureatom sections 1-2, 7]: alpha speed u = 2.09 x 10^9 cm/s; central charge N = 100 unit charges; e = 4.65 x 10^-10 e.s.u.; alpha charge-to-mass ratio E/m = 1.5 x 10^14 e.s.u./g; gold foil t = 4 x 10^-5 cm ("0.00004 cm"); observed "about 1 in 20,000 ... turned through an average angle of 90 degrees" (Geiger and Marsden, 1909). Modern values [memory, high]: alpha mass 6.645 x 10^-24 g, gold Z = 79, density 19.32 g/cm^3, atomic mass 196.97, fastest radium alpha (radium C') 7.69 MeV.

Run it yourself: .venv/bin/python review/Rutherford-Ernest/example.py

Step 1. The alpha's energy. K = (1/2) m u^2 = 0.5 x 6.645 x 10^-24 x (2.09 x 10^9)^2 = 1.45 x 10^-5 erg = 9.06 MeV. Since u/c = 0.070, Newtonian mechanics is fine. Note that 9.06 MeV is higher than any radium alpha; the fastest, radium C', is 7.69 MeV (1.93 x 10^9 cm/s). The 1911 speed was the value then in use and was about 8 percent too high [calc; Arbiter R4].

Step 2. The head-on distance b. At the turning point all kinetic energy is potential energy: (1/2) m u^2 = N e E / b, so b = 2 N e (E/m) / u^2 = 2 x 100 x 4.65 x 10^-10 x 1.5 x 10^14 / (2.09 x 10^9)^2 = 3.19 x 10^-12 cm. The paper prints "about 3.4 x 10^-12 cm", about 6 percent higher; the paper shows no working, so the cause is unknown [read: deep/structureatom section 2]. With modern inputs (Z = 79, K = 7.69 MeV): b = 2 x 79 x 1.44 / 7.69 fm = 29.6 fm = 2.96 x 10^-12 cm. Either way b is about 3,000 times smaller than the atom (10^-8 cm). Units check: N e E / b is (e.s.u.)^2/cm = dyne cm = erg, the unit of (1/2) m u^2.

Step 3. Which paths turn through 90 degrees. From cot(phi/2) = 2p/b, phi = 2 arctan(b/2p):

These are Rutherford's printed values to the decimal [read: deep/structureatom section 3; calc]. A 90-degree turn needs p = b/2, about 1.6 x 10^-12 cm: the alpha has to aim within 1/6000 of the atom's radius of the centre.

Step 4. The fraction beyond 90 degrees. Atoms per cm^3 in gold: n = 19.32/196.97 x 6.022 x 10^23 = 5.91 x 10^22. Atoms per cm^2 of foil: n t = 2.36 x 10^18. The target area per atom is pi p^2 = pi (b/2)^2, so

f(>90 degrees) = (pi/4) n t b^2 cot^2(45 degrees) = (pi/4) x 2.36 x 10^18 x (3.19 x 10^-12)^2 = 1.9 x 10^-5, about 1 in 53,000.

With modern inputs (b = 2.96 x 10^-12): 1.6 x 10^-5, about 1 in 62,000 [calc]. Observed: about 1 in 20,000, or 5 x 10^-5, which is 2.6 to 3 times more. Rutherford did not do this comparison in the paper. It is the same order of magnitude but not a precise test: Geiger and Marsden's figure is a rough "average angle of 90 degrees", and alphas slowed inside the foil scatter more (b grows as 1/u^2) [read: deep/structureatom section 4; Arbiter R4]. The precise tests came later (Step 6).

Step 5. Why many small deflections cannot do it. In J. J. Thomson's 1910 picture the positive charge is spread through the whole atom, so each atom deflects an alpha only a little, and a large angle has to be built up from many small kicks in a random walk. Rutherford gave the rms (root-mean-square) angle after crossing a foil as theta_t = (3 pi b/8) sqrt(pi n t), using the average small deflection of his own central-charge atom, which is three times larger than Thomson's uniform sphere for the same charge, so this is generous to the compound picture [read: deep/structureatom section 6]. With b = 3.19 x 10^-12 cm and n t = 2.36 x 10^18:

theta_t = (3 pi x 3.19 x 10^-12 / 8) x sqrt(pi x 2.36 x 10^18) = 0.0103 rad = 0.59 degrees.

The chance of building up 90 degrees (pi/2 rad) is e^(-(pi/2)^2/theta_t^2) = e^(-23,500), about 10^-10,200 [calc]. Single scattering gives 10^-5. A power law (single close encounters) beats a Gaussian (many small ones) by ten thousand orders of magnitude in the tail. This is the whole logic of the 1911 paper in one number: the rare large deflections must come from single encounters with a tiny, intense centre. Rutherford himself warned that compound and single scattering give the same general form when half the particles are scattered (his comparison with Crowther's beta-ray data), so the large-angle tail is where the theories part [read: deep/structureatom section 7d].

Step 6. The angular law. Equation (5) predicts the count per unit area goes as cosec^4(phi/2). Between 15 and 150 degrees that ratio is cosec^4(7.5 degrees)/cosec^4(75 degrees) = 3445/1.15 = about 3,000. The 1930 book's table (the angle test, made with a radon source and silver and gold foils; it is the Geiger-Marsden 1913 test [memory, high; the book does not name it on that page]) shows gold counts of 132,000 at 15 degrees and 33.1 at 150 degrees, a ratio of about 4,000; silver 105,400 and 22.2, about 4,700 [read: 1930 book p.224, deep/Radiations section 2.1; calc]. So the law holds over a 250,000-fold range of count rate (5 to 150 degrees across both sets), with a 30 to 60 percent excess at the smallest angles in the first set that the book does not comment on (beam width and plural scattering are the likely causes, not stated by the authors) [Arbiter, Claim 4].

Step 7. What the scattering says about the size of the nucleus. The closest distance on a path turned through phi is d = (b/2)(1 + cosec(phi/2)). For 7.5 MeV alphas at 135 degrees:

[read: 1930 book p.235; calc; nuclear radius formula [memory, high]]. The book's numbers reproduce, and its conclusion is worded correctly: an upper bound. Coulomb's law held at those distances because the alphas never got closer than 2.4 to 4.5 nuclear radii.

Now a light nucleus. Aluminium (Z = 13, A = 27) hit head-on by a 7.69 MeV alpha: the target recoils, so the energy available for the approach is the centre-of-mass energy, K x 27/(27 + 4) = 6.70 MeV, and d = 2 x 13 x 1.44 / 6.70 = 5.6 fm. The two nuclei touch at 1.2 x (27^(1/3) + 4^(1/3)) = 5.5 fm [calc]. The alpha reaches the nucleus, so the pure Coulomb law must fail for light targets. That is just what the 1920s experiments found (Claim 6: departures from Coulomb scattering in light elements, and nitrogen disintegrated by alphas), and it is why heavy-element scattering could only ever give an upper bound [calc]. The 1930 book reports: hydrogen hit by 8.2 cm-range alphas scattered at 30 degrees about 100 times more often than Coulomb's law allows (4.3 x 10^-5 against 4.4 x 10^-7; Chadwick and Bieler); helium about 20 times more at 40-50 degrees; aluminium and magnesium fall below the Coulomb value (to about 0.35 at 135 degrees), with Coulomb's law holding only down to about 13 x 10^-13 cm (13 fm) [read]. The authors concluded "a hitherto unknown, though not altogether unsuspected, force"; their 1920s models of it (an oblate-spheroid alpha, an inverse-fourth-power attraction, polarisation) did not survive [read: 1930 book Ch. IX, page images, via the Arbiter QE revision section 5].

The opposite puzzle came from uranium, and it is the cleanest one in the books. Fast alphas scattered from uranium obey Coulomb's law within 5 percent down to 3.9 x 10^-12 cm, so the nucleus is smaller than that. Yet uranium's own alpha leaves with only 4.07 MeV, and classically it must start from where the Coulomb energy equals that: r = 2 Z e^2 / E = 2 x 90 x 1.44 / 4.07 fm = 64 fm = 6.4 x 10^-12 cm (Z = 90 for the daughter, thorium), outside the size the scattering allows [read: 1930 book Ch. XI; calc, example.py Part C]. The book calls this "difficult to reconcile on any theory based on the classical mechanics". Wave mechanics resolved it (Gamow; Gurney and Condon, 1928 [memory, high]): the alpha tunnels through the Coulomb barrier instead of climbing over it. Rutherford's own 1927 "satellite" model of alpha emission is criticised in the same book and was contradicted by his group's own null test (Herszfinkiel and Wertenstein) [Arbiter QE revision section 10].

Step 8 (Part B in example.py). The 1904 heat budget. Rutherford's 1904 alpha energy was 5.9 x 10^-6 erg. Compute the energy as E = (1/2) q v^2/(q/m) and compare his inputs with the true values factor by factor: charge (one e of 3.39 x 10^-10 against two of 4.80 x 10^-10) 0.353; speed squared (2.5 x 10^9 against 1.92 x 10^9) 1.70; q/m (true 4.82 x 10^3 against his 6 x 10^3 emu/g) 0.80. Product 0.48, so the true energy of the alpha he measured is 5.9 x 10^-6 / 0.48 = 1.23 x 10^-5 erg = 7.7 MeV: radium C'. And the modern four-emitter alpha heat (Section 3) is 124 cal/h per gram, the whole observed output, with 80 percent from the emanation and its products (Rutherford and Barnes: about 75 percent) [Arbiter R1; calc]. The "missing" energy of 1904 was bookkeeping.

What the example shows. The nuclear atom was not a guess fitted to one surprising number. It made a quantitative prediction from measured inputs (to within a factor of 3 for a crude observation), it was the only picture that could produce the large-angle tail at all (Step 5), its angular law was confirmed over a huge range (Step 6), and it was honest about its own limit (Step 7). Its errors were in the inputs (alpha speed, central charge), and they were found and fixed within 15 years, mostly by the same group.

6. Evidence table

7. What a learner should take away

What is solid. The exponential law of radioactive decay and the parent-daughter equations; the internal, nuclear origin of radioactive heat, with the right size (3.6 x 10^9 cal per gram of radium, a million times chemistry); alpha particles as helium nuclei; the small, massive, positive nucleus and the cosec^4 scattering law; radioactive heat as the reason Kelvin's age of the earth was wrong; the first artificial transmutation. Every one of these was checked numerically by the reviewers and every printed derivation in the 1911 paper closes [Arbiter].

What was wrong, and why the mistakes are instructive.

• Input errors that cancel. The 1904 alpha energy was off by a factor of two because of four separate wrong inputs, two pushing each way. When a result disagrees with an energy budget, check each input before inventing a new energy source. Rutherford himself reached for "rearrangement energy"; the correct answer was arithmetic [Arbiter R1].
• An average is not a distribution. The 1905 "270 million tons of radium" is a whole-earth average; put crustal concentrations everywhere and you get 200 times too much heat. The 1913 book noticed this and moved the radioactivity into a thin shell (Strutt). The same lesson applies to any claim that sounds like "X is enough to explain Y": ask where X is.
• Scaling is not a calculation. "A hundredfold" longer life for the sun was a scaling of Helmholtz's number; the direct estimate gives 2.6 x 10^9 years with the wrong mechanism, and fusion gives 40 times more energy per gram.
• A replicated anomaly can still be an artifact. The Curie-Danne temperature effect was seen by two laboratories and accepted in print by Rutherford before Bronson showed it was a mixed sample. That is a good model for how to treat any reported change in decay rate: ask what else could have changed between "before" and "after".
• Know which way the classical picture fails. Uranium scattering said the nucleus is smaller than 3.9 x 10^-12 cm; uranium's alpha energy said, classically, it must leave from 6.4 x 10^-12 cm. Both measurements were right. The contradiction was the sign that classical mechanics does not apply inside the nucleus, and wave-mechanical tunnelling resolved it (Section 5, Step 7).
• Saving a law by postulating something unseen. The continuous beta spectrum looked like a failure of energy conservation; Bohr was ready to give it up [memory, high]. Pauli instead postulated an undetected particle (1930), and it was found 26 years later. That is the clearest case in this project where an accepted law was kept by postulating something unseen and the postulate was then confirmed; read it beside cases where the postulated thing was never found.
• How claims of easy transmutation were rejected. The 1920s claims of gold from mercury and helium from hydrogen in palladium were answered mostly by repeating the experiments, by atomic-weight and isotope checks, and by the claimants' own withdrawals; the hidden source in the helium claims was helium from the air absorbed by glass. Any new claim of this kind has to meet the same controls.
• Hedges matter. The 1911 paper did not claim a positive "nucleus"; it claimed a small central charge of unknown sign. The 1919 paper did not claim protons from nitrogen; it said the evidence could not yet decide. Later summaries (including this project's own first claims list) stated more than the sources did, and the reviewers corrected them.

What is genuinely open, or at least untested in the record.

1. Gravity and decay rates: Rutherford named gravity in 1904 (first edition of Radio-Activity, sec. 206) as the one force "not under our control", and the Rutherford-Compton method of 1919 was interrupted by the war as far as the record shows. Against that, the 1930 book, which Rutherford co-wrote, says gravitational fields were "looked for ... without a positive result", with no reference. The two statements do not match, and no source says which is right: the gravity looking may have been done, found nothing and not been written up, or never been carried out. The deciding records are the body of the Nature letter (104, 412), Cambridge MS Add.7653, and whatever Rutherford, Chadwick or Ellis recorded behind the 1930 sentence [Arbiter, Absence of evidence 1; QE revision section 7]. A related 1904 question, whether the parts of a broken-up atom weigh the same as the atom, rests on weight changes of radium reported by Heydweiller and Dorn, "not confirmed", and a null by Forch.
2. Bumstead's 1906 lead-zinc X-ray heating factor of two: the physicist's suggestion (secondary and fluorescent radiation) has not been shown to work in the right direction, as the contrarian notes; Bumstead himself raised it and argued it does not work; and his radiometer check rules out a simple ionisation-calibration artifact. No replication or refutation was found. A calorimetric repeat with monochromatic X-rays would settle it [Arbiter, Claim 7d and QE revision section 8].
3. Residues that the 1913 and 1930 books leave unexplained: Hoffmann's and Ziegert's weak alphas from platinum, brass and zinc, "further information must be awaited"; Joly's X haloes and Iimori and Yoshimura's Z haloes, whose "exact origin is unknown" (Ziegert suggested a link between the two), and the halo radii for uranium that are "not easy to account for"; Poole's mineral heat 1.8 times the calculated value. No later resolution was searched for in this review [Arbiter QE revision sections 2 and 8].
4. Curie and Danne's response to Bronson, not found; the Vienna-Cambridge disagreement over which light elements disintegrate (the records that would settle it, Kirsch and Pettersson 1926 and Chadwick 1926, are not read here); Riding and Baly's helium and neon from nitrogen, which the 1930 book does not answer specifically; the small-angle excess in the 1930 angle table.
5. Modern claims of solar-correlated decay-rate variation: contested in the literature and not evaluated here.
6. The "detonator" itself. Trigger by radiation: tested from 1903 to 1928 and not found, at limits as tight as 1 in 8 million (these tests probe only the radiations then available and alpha and beta emitters). Trigger by "some unknown external force": not testable as stated, because no force is named. The idea survives in the other form (projectile-induced reactions). Readers coming from the Moray and Le Bon projects should note that Rutherford's own position was always "energy from the atom, trigger conjectural", and that he argued against external energy sources in the same address Le Bon quoted.

Dewar's 1903 liquid-hydrogen report, listed here as open in the first version of this primer, is closed: Curie and Dewar measured it again in 1904 and found no excess, though in the same laboratory.

What Rutherford and others expected of atomic energy. In 1921 Richardson told the British Association that Rutherford's work made artificial transmutation "an established fact. If atomic energy could be intensified and controlled, we would have at our disposal an illimitable supply of power" [web: https://paperspast.natlib.govt.nz/newspapers/CHP19210913.2.57; OCR noisy]. In 1933 Rutherford told the same body that hopes of releasing energy by splitting atoms had proved "moonshine" (Levin Daily Chronicle, 13 Sept 1933, a cable report of the lecture) [web: Papers Past, via sites.md]. Both are expectations, not measurements, from an interested party (Richardson) and from the inventor himself; the 1933 remark is known here from one newspaper cable, not from the lecture text. Fission was found five years later (1938) [memory, high].

8. Check yourself

1. Thorium X has a half-life of about 4 days. What is lambda in units of per hour, and what fraction of a freshly separated sample of thorium X is left after 12 days?
2. In Step 4, what happens to the predicted fraction scattered beyond 90 degrees if (a) the foil is twice as thick, (b) the alpha energy is halved, (c) the target is silver (Z = 47) instead of gold, keeping n t the same?
3. Why does the closest-approach distance at 135 degrees give only an upper bound on the nuclear radius, while scattering from aluminium can give a lower bound on where Coulomb's law fails?
4. Rutherford's 1904 alphas seemed to supply only 40-71 of about 100 cal/h. Name the four input errors and say which way each pushed the answer.
5. A writer quotes Rutherford as "in doubt whether the energy is borrowed from the atom itself or from external sources". What else should be quoted beside it, and why?

9. Further reading

Mainstream. - K. S. Krane, Introductory Nuclear Physics (Wiley, 1988): radioactive decay, alpha decay, nuclear sizes, Rutherford scattering. - J. R. Taylor, Classical Mechanics (University Science Books, 2005): central forces and the classical derivation of the Rutherford cross-section. - A. Pais, Inward Bound: Of Matter and Forces in the Physical World (Oxford, 1986): history of radioactivity and the nucleus [memory, high]. - L. Badash (ed.), Rutherford and Boltwood: Letters on Radioactivity (Yale, 1969): the private correspondence behind the decay chains [via sites.md]. - J. Campbell, Rutherford: Scientist Supreme (AAS Publications, 1999), and his online bibliography [web: https://www.rutherford.org.nz/bibliography.htm]. - H. A. Bethe, "Energy production in stars", Phys. Rev. 55, 434 (1939): how the sun is really powered [memory, high]. - N. Todd, "A brief history of Lord Rutherford's radium" (2014) [web: https://pmc.ncbi.nlm.nih.gov/articles/PMC4123660/].

Rutherford's own key works (in the project folder unless marked). - With F. Soddy, the 1902-03 Phil. Mag. papers on "The cause and nature of radioactivity" (not in the folder; in the Collected Papers, ed. Chadwick, 1962-65). - Radio-Activity (Cambridge, 1904; sec. 206 has the detonator and gravity sentences; the 2nd ed. 1905, sec. 270, repeats them [web: https://archive.org/details/radioactivity00ruth]). - "The Succession of Changes in Radioactive Bodies", Bakerian Lecture, Phil. Trans. A 204 (1905) 169-219 (saved in research/files/). - "Present Problems in Radioactivity", St Louis address 1904 [web: https://en.wikisource.org/wiki/Popular_Science_Monthly/Volume_67/May_1905/Present_Problems_in_Radioactivity]. - Radioactive Transformations (Silliman Lectures, Yale, 1906). - "The Scattering of alpha and beta Particles by Matter and the Structure of the Atom", Phil. Mag. 21, 669 (1911). - Radioactive Substances and Their Radiations (Cambridge, 1913). - "Atomic Projectiles and their Collisions with Light Atoms", Science 50 (1919) 467-473 (saved in research/files/). - "Some Cosmical Aspects of Radioactivity", J. Roy. Astr. Soc. Canada (1907) 145-165: radioactivity and the ages of the earth and sun (not in the folder; listed in Campbell's bibliography) [web: https://www.rutherford.org.nz/bibliography.htm, via sites.md]. - With J. Chadwick and C. D. Ellis, Radiations from Radioactive Substances (Cambridge, 1930). - The Natural and Artificial Disintegration of the Elements, Franklin Institute centenary address (September 1924): the helium nucleus ("four or five of the swiftest alpha particles would be necessary to effect its disruption") and the early "satellite" picture of alpha emission (not in the folder) [web: https://archive.org/details/naturalartificia0000ruth, via sites.md].

Related work kept in the record. Bronson, Am. J. Sci. 20 (1905) 60-64 (temperature effect explained); Bumstead, Am. J. Sci. 21 (1906) 1-24 (X-ray heating); Gates, Phys. Rev. 16 (1903) and 18 (1904) (heat and excited activity; Le Bon's quinine effect; Le Bon disputed her explanation) [web: http://cwp.library.ucla.edu/articles/gates/gates1.html]; Le Bon, The Evolution of Matter (1905, English 1907) [web: https://www.rexresearch.com/lebonmat/lebonmat.htm], an interested party; Lucien Poincare, The New Physics and Its Evolution (1907) [web: https://www.gutenberg.org/ebooks/15207]; Ellis and Wooster, Proc. Roy. Soc. A 117 (1927), and Meitner and Orthmann, Zeit. f. Phys. 60, 143 (1930) (heat of beta decay) [Ellis-Wooster reference: memory, medium; Meitner-Orthmann: Arbiter]; Paneth and Peters (1926) and Paneth's later retraction, cited in the 1930 book Sec. 74; Poole, Phil. Mag. 19, 314 (1910) (mineral heat); Ziegert, Zeit. f. Phys. 46, 668 (1927) (residual alphas); Kirsch and Pettersson, Atomzertruemmerung (1926), and Chadwick, Phil. Mag. 2, 1056 (1926) (the Vienna-Cambridge dispute) [Arbiter, QE revision section 13]; G. T. Emery, "Perturbation of Nuclear Decay Rates", Annu. Rev. Nucl. Sci. 22 (1972) 165-202 (located, not read).

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Answers

1. lambda = ln 2 / 96 h = 0.00722 per hour. Twelve days is three half-lives, so (1/2)^3 = 1/8 is left (check: e^(-0.00722 x 288) = e^(-2.08) = 0.125).
2. The fraction is (pi/4) n t b^2 cot^2(phi/2) with b proportional to Z/K. (a) Doubles (proportional to t). (b) b doubles, so the fraction goes up four times (proportional to 1/K^2, the "fourth power of the velocity" of Rutherford's prediction (4)). (c) Proportional to Z^2: (47/79)^2 = 0.35 of the gold value. Note: it goes as the square of the central charge, not the charge itself; the 1911 transcription's "magnitude of central charge" is loose, and Rutherford states the square on p.16.
3. For gold the alphas stopped 2.4 to 4.5 nuclear radii away, so seeing pure Coulomb scattering only shows the nucleus is no bigger than the closest distance reached; it says nothing about how much smaller. For aluminium the closest approach (about 5.6 fm) is at the touching distance, so a departure from the cosec^4 law there marks roughly where the nuclear force begins, which gives a size estimate rather than only a bound.
4. Alpha charge taken as e instead of 2e (made the energy too low); e itself 3.39 x 10^-10 instead of 4.80 x 10^-10 e.s.u. (too low); speed 2.5 x 10^9 instead of 1.92 x 10^9 cm/s (too high); q/m 6 x 10^3 instead of 4.82 x 10^3 emu/g (too low, because E = (1/2) q v^2/(q/m)). Net factor 0.48: his 3.7 MeV becomes 7.7 MeV.
5. Quote the English wording of the same sentence ("whether the view is taken that the energy is derived from the atom itself or from external sources"), the argument against external sources that follows it in the same address (heat inside an ice calorimeter, unchanged under several inches of lead), and the detonator passage, which says that even if an outside force triggers the break-up, the energy comes from the atom. Le Bon, who claimed priority for related ideas, quoted the first sentence alone; the full context shows Rutherford held the energy to be internal.