Becquerel-Henri: Primer

Source gap (partial, carried from Steps 1 and 3). The folder holds three short documents: Becquerel's own article "On radio-activity, a new property of matter" (Nature 63, 21 February 1901, pp. 396-398; scanned; the OCR text is readable prose but garbles exponents, so every number was read from page images), R. K. Duncan's popular piece in Harper's (August 1902), and J. J. Thomson's "Becquerel Rays" in Harper's (1903). Becquerel's 1896 Comptes Rendus notes and his 1903 Nobel lecture are not in the folder, so the 1896 discovery story rests on his 1901 retelling and on the popular accounts. The numbers used in the worked example below were checked against the page image of Nature p.398 [read: page image p.398]. This revision also uses two deep reads made after the quality check (in deep/): Nature pp.396-398 from page images, and Thomson pp.291-293.

Framing. This is not a suppressed or fringe project. It is the founding record of radioactivity, and it contains three instructive wrong turns made by first-rate scientists: Becquerel's claim that the energy books balance, Marie Curie's passing suggestion that radium draws energy "from space", and Thomson's guess that ordinary metals become radioactive through layers of air ions. Nothing in the folder documents resistance or suppression [Arbiter].

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1. The idea in one paragraph

In 1896 Henri Becquerel found that uranium salts, kept in the dark, fog a wrapped photographic plate and make the air around them conduct electricity. Nothing has to light them up or heat them first; the effect simply comes out of the uranium, and keeps coming out year after year. By 1901 he and others had shown that this "radio-activity" belongs to certain elements (uranium, thorium, and the new radium and polonium found by the Curies), that part of the radiation is bent by a magnet in the same way as the electron beams in a cathode-ray tube, and that another part is not bent. The puzzle was energy: radium keeps pouring out radiation without any visible fuel. Becquerel estimated how much matter the electrons carry away, found it tiny, and concluded there was "no contradiction" with energy conservation. That conclusion was wrong, and finding out why it was wrong led straight to the nucleus and to the idea that atoms transform into other atoms.

2. What you need to know first

Charged particles in a magnetic field. A particle of charge q moving at speed v perpendicular to a uniform magnetic field B feels a force qvB at right angles to its motion, so it travels in a circle. Setting that force equal to the mass times the centripetal acceleration, m v^2 / r = q v B, gives r = m v / (q B). The product B r (called "magnetic rigidity", written H rho in old papers) therefore equals momentum divided by charge. Measuring the circle radius in a known field tells you p/q. Read: Griffiths, Introduction to Electrodynamics, the chapter on magnetostatics (Lorentz force law and cyclotron motion).

Combining electric and magnetic deflection. An electric field bends a beam by an amount that depends on the particle's kinetic energy per charge; a magnetic field bends it by an amount that depends on momentum per charge. Two different measurements, two unknowns (v and q/m): solve for both. This is exactly how J. J. Thomson measured the electron's charge-to-mass ratio in 1897 and how Becquerel did it for radium rays in 1900. Read: any intro modern physics text, e.g. Serway, Moses and Moyer, Modern Physics, the chapter on the electron and Thomson's e/m experiment.

CGS electromagnetic units (emu). Old papers use centimetres, grams and seconds, with magnetic field in gauss and charge in "abcoulombs" (emu of charge). One emu of charge is 10 coulombs; one emu of current is 10 amperes. In these units the force on a moving charge is simply q v B (no factor of c), so r = m v/(q B) holds as written. Energy is in ergs: 1 erg = 10^-7 joule. The electron's charge-to-mass ratio is 1.76 x 10^11 C/kg = 1.76 x 10^7 emu/g [memory: high]. Read: the appendix on units in Griffiths, or Jackson's Classical Electrodynamics appendix on units if you want the full story.

Relativistic momentum. At speeds that are a sizeable fraction of the speed of light c, momentum is p = gamma m v, where gamma = 1/sqrt(1 - v^2/c^2) and m is the rest mass. A magnet measures p, so if you use the Newtonian p = m v at half the speed of light you get the wrong v. It is convenient to write beta = v/c; then p/(m c) = gamma beta, and beta = (gamma beta)/sqrt(1 + (gamma beta)^2). Kinetic energy is (gamma - 1) m c^2. Read: Serway, Moses and Moyer, the special relativity chapters; or Taylor and Wheeler, Spacetime Physics.

Exponential decay. If each atom has a fixed probability lambda per second of decaying, then a sample of N atoms loses lambda N atoms per second: dN/dt = -lambda N, so N(t) = N_0 e^(-lambda t). The half-life is T_1/2 = ln 2 / lambda. The number of decays per second, lambda N, is called the activity. This is an ordinary first-order differential equation; nothing more is needed. Read: Krane, Introductory Nuclear Physics, the chapter on radioactive decay; or the radioactivity chapter of any intro modern physics text.

Energy from mass. In a nuclear decay the products weigh slightly less than the parent; the missing mass Delta m appears as kinetic energy Q = Delta m c^2. Per atom, nuclear Q values are a few million electron volts (MeV), about a million times larger than chemical reaction energies (a few eV). Read: Krane, chapters on nuclear masses and alpha decay.

3. The textbook view

What radioactivity is. Certain nuclei are unstable and change spontaneously into other nuclei, emitting:

• alpha particles: helium-4 nuclei, charge +2e, mass about 4 atomic mass units. Heavy and slow (a few percent of c), so a magnet that bends electrons strongly bends them only slightly, and in the opposite direction. They are stopped by a sheet of paper.
• beta particles: electrons (charge -e, mass m_e = 9.11 x 10^-28 g), emitted with a spread of speeds, often above half of c. Bent strongly by a magnet; stopped by a few millimetres of aluminium.
• gamma rays: high-energy photons (electromagnetic radiation, like X-rays but from the nucleus). Not bent at all; very penetrating.

[memory: high for all three identifications.] The 1901-03 sources were already circling these answers. Becquerel separated three kinds of ray by penetration (Section 4). Thomson noted in 1903 that "the minerals in which helium occurs invariably contain radio-active substances" and that "helium may be very closely connected with the gaseous emanation" [read: Thomson, Harper's pp.292-293]. That was a guess with no experiment behind it. That alpha particles are helium nuclei is later knowledge [memory: high].

Key equations (symbols defined in Section 2):

• Rigidity: B r = p / q. Nonrelativistic: v = (q/m) B r. Relativistic: gamma beta = (q/m) B r / c.
• Decay: dN/dt = -lambda N; activity A = lambda N; T_1/2 = ln 2 / lambda.
• Power released by a sample: P = A x Q, where Q is the energy per decay.

Why it never "runs out" on a human timescale. Uranium-238 has a half-life of 4.5 x 10^9 years [memory: high], so its activity is constant over any laboratory lifetime. Radium-226 has a half-life of about 1600 years [web: https://www.chemlin.org/isotope/radium-226, via arbiter]. It is not inexhaustible, merely slow. Polonium-210 has a half-life of 138 days [memory: high], and Becquerel's article itself reports that polonium's activity weakens with time [read: page image p.397].

Where the energy comes from. Each decay releases a fixed Q. For radium-226, Q = 4.87 MeV [web: chemlin, via arbiter]. One gram contains 6.02 x 10^23 / 226 = 2.7 x 10^21 atoms, of which 3.7 x 10^10 decay per second (this number defined the old unit, the curie). Multiply and you get the heat output (Section 5, Step 5).

Decay rates are insensitive to the environment. Alpha and beta decay rates do not change measurably with temperature, chemical form or pressure. Curie and Dewar found radium's heat output unchanged at liquid-hydrogen temperature [web: https://history.aip.org/exhibits/curie/article.htm; https://www.nobelprize.org/uploads/2018/06/pierre-curie-lecture.pdf].

4. Where this researcher (and his contemporaries) depart

Becquerel is the textbook here; most of what he reported is now standard. The departures are specific inferences, set beside the modern view.

The exact point of departure that matters most is the energy row. Becquerel's arithmetic is correct; it is the physical premise (that the escaping electron mass measures the energy flow) that fails.

5. Worked example: does Becquerel's energy argument hold?

Everything below is reproduced in example.py in this folder. Inputs from Nature p.398 [read: page image p.398]:

• H rho = 1600 gauss cm (the rigidity of the particular "simple radiation" he measured)
• e/m = 10^7 emu/g (he prints the inverse, m/e = 10^-7)
• Curie screen current j = 4 x 10^-13 emu of charge per cm^2 of radium surface per second (= 4 x 10^-12 A/cm^2)

Step 1: speed, Newtonian (Becquerel's method). v = (e/m)(H rho) = 10^7 x 1600 = 1.6 x 10^10 cm/s. With c = 3.00 x 10^10 cm/s, v/c = 0.534. This matches his "1.6 x 10^10, about one-half of that of light".

Step 2: the relativistic correction. The magnet measures momentum, so gamma beta = (e/m)(H rho)/c. - With his e/m: gamma beta = 0.534, so beta = 0.534 / sqrt(1 + 0.534^2) = 0.534 / 1.134 = 0.471. - With the modern e/m = 1.76 x 10^7: gamma beta = 0.939, beta = 0.939 / sqrt(1.882) = 0.684.

So the same rigidity actually means an electron at 0.68 c. Two errors (a Newtonian formula and a low e/m) partly cancel in his 0.53. Thomson's later "over 120,000 miles per second" (0.64 c) for the fastest radium rays [read: Thomson] is a different, faster component, not a contradiction.

Step 3: mass carried away. Mass per second = charge per second divided by charge per mass: j / (e/m) = 4 x 10^-13 / 10^7 = 4 x 10^-20 g per cm^2 per second. Over 10^9 years (3.156 x 10^16 s): 4 x 10^-20 x 3.156 x 10^16 = 1.26 x 10^-3 g = 1.26 mg per cm^2. Becquerel says "about one milligramme in one thousand million years". Correct. Thomson's "one-thousandth of a milligram in a million years" [read: Thomson] is the same number.

Step 4: energy carried by those electrons. Kinetic energy flux, Newtonian (as Thomson did it): (1/2)(4 x 10^-20 g/s/cm^2)(1.6 x 10^10 cm/s)^2 = 5.1 erg/s/cm^2 = 5.1 x 10^-7 W per cm^2. Over a million years (3.156 x 10^13 s): 1.6 x 10^14 erg = 1.6 x 10^7 J per cm^2. Melting ice takes 334 J/g, so this melts 4.8 x 10^4 g; at 0.917 g/cm^3 that is a column 1 cm^2 across and 528 m (0.33 mile) tall. Thomson: "more than a quarter of a mile" [read: Thomson]. Correct. Relativistic check: with gamma = sqrt(1 + 0.534^2) = 1.134, the kinetic energy (gamma - 1) m c^2 gives 4.8 erg/s/cm^2, 6% less. The Newtonian figure is fine for this purpose.

Step 5: what radium actually emits. Power = (decays per second) x (energy per decay). - Atoms in 1 g of Ra-226: N = 6.022 x 10^23 / 226 = 2.66 x 10^21. - Decay constant: lambda = ln 2 / (1600 x 3.156 x 10^7 s) = 1.37 x 10^-11 per second. - Activity: lambda N = 3.66 x 10^10 decays per second. - Ra-226 alone, Q = 4.87 MeV = 7.80 x 10^-6 erg: power = 2.9 x 10^5 erg/s = 0.029 W/g. - Within a few weeks the radium's short-lived daughters (radon-222, polonium-218, polonium-214) reach equilibrium and each adds its own alpha decay (Q about 5.59, 6.11 and 7.83 MeV [memory: high]). Total alpha energy per radium decay: 24.4 MeV, giving 0.14 W/g. - Measured by Curie and Laborde in 1903: about 100 calories per hour per gram = 0.12 W/g [web: https://history.aip.org/exhibits/curie/article.htm]. Agreement within about 20%, which is as good as this rough accounting allows (part of the beta energy goes to neutrinos, and their samples were impure salts).

Step 6: compare. A gram of radium salt spread thinly has an emitting surface of order 1 cm^2 (a rough assumption; the conclusion does not depend on it to better than a factor of 10). Then

measured output / Becquerel's beta estimate = 0.12 W / 5.1 x 10^-7 W = about 2 x 10^5.

Caution: this ratio is an upper-end figure. It divides the heat of pure radium (Curie and Laborde, 1903) by a screen current measured on a 1900 preparation whose purity Becquerel does not state. The same article says the Curies' products of that period were only "several thousand times" as active as uranium [read: page image p.397]. Pure radium is about 3 x 10^6 times as active per gram (3.7 x 10^10 against about 1.2 x 10^4 decays per second per gram [memory: high, not web-checked]). So the fair statement is "several orders of magnitude, up to about 10^5 for pure radium" [Arbiter, QE revision].

A sounder argument that does not need the purity. Compare energy per decay. With its daughters in equilibrium, one radium decay releases about 24 MeV in alpha particles (Step 5). The two beta decays in the chain (lead-214 and bismuth-214) give their electrons well under 1 MeV each on average, so under 2 MeV per radium decay in total, because neutrinos carry off the rest of the beta energy unseen [memory: high]. So beta electrons carry under a tenth of the energy, whatever the sample's purity. The alphas, which carry the rest, stop within a few hundredths of a millimetre of solid and heat the source, where a screen outside cannot see them.

What the example shows. Becquerel's arithmetic is right, and his conclusion is wrong by several orders of magnitude. The screen caught only the beta electrons that escaped the source and reached the screen; alpha particles (not yet identified) carry most of the energy and are absorbed in the radium itself, heating it. More deeply, the energy does not come from the emitted mass being "spent" like perfume evaporating; it comes from the nuclear mass difference Delta m c^2. A tiny mass loss can carry an enormous energy, which is exactly the opposite of the reassurance he drew.

6. Evidence table

Verdicts are the arbiter's final verdicts [Arbiter].

7. What a learner should take away

Solid. Spontaneous emission, its independence from light and temperature, the identification of beta rays as electrons by combined electric and magnetic deflection, and induced activity are all established, and Becquerel's 1901 numbers are internally consistent to within a few percent of what he says.

Wrong, and why that is instructive. - The energy argument. Correct arithmetic built on the wrong premise. The lesson: before concluding "the books balance", ask whether your measurement captures the whole energy flow. Becquerel measured one channel (escaping betas) and treated it as the total. Per decay, the betas carry under 2 MeV against about 24 MeV of alpha energy; the alphas, absorbed inside the sample, were invisible to his screen but not to a calorimeter. The calorimeter (Curie-Laborde) was the decisive instrument. - Newtonian formulas near c/2. An 18% error in gamma at v = 0.53 c. Becquerel can be excused; relativity came in 1905. The habit to build: check v/c before trusting p = m v. - Thomson's ion layers. A sensible first step (a negative body attracts positive ions) followed by a quantitative leap nobody checked. One line of electrostatics (field from a sheet of charge) rules it out. The lesson: always attach an order-of-magnitude number to a proposed mechanism. - Instruments ahead of understanding. The 1901 article already records radium salts growing more active after preparation and polonium weakening, and the 1903 article records uranium regrowing its activity after twelve months. These are ingrowth and decay, seen before anyone could explain them. Rutherford and Soddy's transformation theory explained them. The observations were sound even while the interpretations were wrong. - "Energy from space." Worth noting for an alternative-science reader: this was a real suggestion by Marie Curie, reported in a popular magazine. It was not refuted by experiment; it was displaced, because the transformation theory predicted things (half-lives, decay and regrowth curves, the heat output from Q) that were then measured. That is how an unneeded hypothesis usually dies.

Genuinely open (historically, not physically). Which 1896 data led Becquerel to drop his early idea that the rays could be refracted and polarised (the 1896 Comptes Rendus notes are needed); what Thomson's "recent experiments" on induced activity actually were (he gives no data); whether Curie's space-energy idea appears in her own writing beyond Duncan's retelling [Arbiter].

8. Check yourself

1. Becquerel's electrons had H rho = 1600 gauss cm. What speed would an alpha particle (charge +2e, mass 7300 electron masses) have at the same rigidity, nonrelativistically? Why did this make alpha rays look "non-deviable" in his magnets?
2. A 1 g sample of pure, freshly purified Ra-226 is sealed in a box. Roughly how does its heat output change over the first two months, and why?
3. Using numbers from Section 5, how much mass (via E = Delta m c^2) does 1 g of radium lose per year as heat at 0.12 W? Compare it with the beta-electron mass loss per cm^2 per year.
4. Why does Becquerel's Newtonian formula with e/m = 10^7 give a speed (0.53 c) closer to the truth (0.68 c) than the relativistic formula with the same e/m (0.47 c)?
5. Thomson's ion-layer idea: a sheet of ions with charge density 10^9 elementary charges per m^3 and thickness d produces a field of about (n e d)/epsilon_0. How thick must the layer be to reach 10^9 V/m? (epsilon_0 = 8.85 x 10^-12 F/m, e = 1.6 x 10^-19 C.)

9. Further reading

Mainstream: - Kenneth Krane, Introductory Nuclear Physics (Wiley, 1988): decay law, alpha and beta decay, Q values. - Serway, Moses and Moyer, Modern Physics: Thomson's e/m experiment, special relativity, intro nuclear physics. - Abraham Pais, Inward Bound (Oxford, 1986): the history of radioactivity and the nucleus, including Becquerel, the Curies and Rutherford-Soddy [memory: high]. - AIP history exhibits: Marie Curie (https://history.aip.org/exhibits/curie/article.htm) and Rutherford (https://history.aip.org/exhibits/rutherford/sections/exploring-radioactivity.html) [web]. - Pierre Curie, Nobel lecture 1905 (https://www.nobelprize.org/uploads/2018/06/pierre-curie-lecture.pdf) [web].

Researcher and contemporaries: - H. Becquerel, "On radio-activity, a new property of matter", Nature 63, pp. 396-398, 21 Feb 1901 (in this folder). - J. J. Thomson, "Becquerel Rays", Harper's Monthly, 1903 (in this folder). - R. K. Duncan, "Radio-activity: a new property of matter", Harper's, August 1902 (in this folder; the folder's file name attributes it to Becquerel, but the text is Duncan's [Arbiter]). - To obtain: Becquerel's 1896 notes in Comptes Rendus (vol. 122) and his 1903 Nobel lecture (nobelprize.org) [memory: high that both exist; volume number medium].

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Answers

1. v = (q/m) H rho. For the alpha, q/m = 2e/(7300 m_e) = (2/7300) x 1.76 x 10^7 = 4.8 x 10^3 emu/g. Then v = 4.8 x 10^3 x 1600 = 7.7 x 10^6 cm/s, far below real alpha speeds (about 1.5 x 10^9 cm/s). Put the other way: a real alpha at 1.5 x 10^9 cm/s has H rho = v/(q/m) = 3 x 10^5 gauss cm, about 200 times the electron's 1600. In a magnet that bends the electrons into a few-centimetre circle, the alpha's circle is metres across, so it looks unbent. Rutherford needed much stronger fields to deflect it (1903).
2. It rises from about 0.03 W to about 0.14 W. Radon-222 (half-life 3.8 days [memory: high]) grows in, and with it the short-lived polonium-218 and polonium-214 daughters; after about a month (several radon half-lives) they are in equilibrium, each decaying as often as the radium, so the alpha energy per radium decay rises from 4.87 to about 24 MeV. The physicist's review cites this rise [web: chemlin]. The 1901 source already records it, unexplained: the Curies and Giesel saw that "the salts of radium increase spontaneously in activity for some time after their preparation" [read: page image p.397].
3. Energy per year: 0.12 W x 3.156 x 10^7 s = 3.8 x 10^6 J. Delta m = E/c^2 = 3.8 x 10^6 / (9 x 10^16) = 4.2 x 10^-11 kg = 4.2 x 10^-8 g per year. Beta-electron mass loss: 4 x 10^-20 g/s x 3.156 x 10^7 s = 1.3 x 10^-12 g per cm^2 per year. The mass-energy loss is about 30,000 times larger than the electron mass carried off: the real "fuel" is invisible in Becquerel's accounting.
4. Two errors cancel. The Newtonian formula overstates v for a given momentum (it should be divided by gamma), while e/m = 10^7 understates v by a factor of 1.76. The net effect lands closer to the true speed than either correction alone would suggest. Agreement with the right answer is not evidence that the method is right.
5. Field per metre of thickness: n e / epsilon_0 = 10^9 x 1.6 x 10^-19 / 8.85 x 10^-12 = 18 V/m per metre. To reach 10^9 V/m: d = 10^9 / 18 = 5.5 x 10^7 m, about four Earth diameters. No such layer can exist in air, so the mechanism fails by many orders of magnitude [Arbiter recomputation].