arash11/wikipedia-physics-corpus · Datasets at Hugging Face

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The beta spectrum, or distribution of energy values for the beta particles, is continuous. The total energy of the decay process is divided between the electron, the antineutrino, and the recoiling nuclide. In the figure to the right, an example of an electron with 0.40 MeV energy from the beta decay of Bi is shown. In this example, the total decay energy is 1.16 MeV, so the antineutrino has the remaining energy: 1.16 MeV − 0.40 MeV = 0.76 MeV. An electron at the far right of the curve would have the maximum possible kinetic energy, leaving the energy of the neutrino to be only its small rest mass.	605	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	900	10	2,024	8	10	31	2	0
Radioactivity was discovered in 1896 by Henri Becquerel in uranium, and subsequently observed by Marie and Pierre Curie in thorium and in the new elements polonium and radium. In 1899, Ernest Rutherford separated radioactive emissions into two types: alpha and beta (now beta minus), based on penetration of objects and ability to cause ionization. Alpha rays could be stopped by thin sheets of paper or aluminium, whereas beta rays could penetrate several millimetres of aluminium. In 1900, Paul Villard identified a still more penetrating type of radiation, which Rutherford identified as a fundamentally new type in 1903 and termed gamma rays. Alpha, beta, and gamma are the first three letters of the Greek alphabet.	720	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	901	11	2,024	8	10	31	2	0
In 1900, Becquerel measured the mass-to-charge ratio (m / e) for beta particles by the method of J.J. Thomson used to study cathode rays and identify the electron. He found that m / e for a beta particle is the same as for Thomson's electron, and therefore suggested that the beta particle is in fact an electron.	313	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	902	12	2,024	8	10	31	2	0
In 1901, Rutherford and Frederick Soddy showed that alpha and beta radioactivity involves the transmutation of atoms into atoms of other chemical elements. In 1913, after the products of more radioactive decays were known, Soddy and Kazimierz Fajans independently proposed their radioactive displacement law, which states that beta (i.e., β) emission from one element produces another element one place to the right in the periodic table, while alpha emission produces an element two places to the left.	503	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	903	13	2,024	8	10	31	2	0
The study of beta decay provided the first physical evidence for the existence of the neutrino. In both alpha and gamma decay, the resulting alpha or gamma particle has a narrow energy distribution, since the particle carries the energy from the difference between the initial and final nuclear states. However, the kinetic energy distribution, or spectrum, of beta particles measured by Lise Meitner and Otto Hahn in 1911 and by Jean Danysz in 1913 showed multiple lines on a diffuse background. These measurements offered the first hint that beta particles have a continuous spectrum. In 1914, James Chadwick used a magnetic spectrometer with one of Hans Geiger's new counters to make more accurate measurements which showed that the spectrum was continuous. The distribution of beta particle energies was in apparent contradiction to the law of conservation of energy. If beta decay were simply electron emission as assumed at the time, then the energy of the emitted electron should have a particular, well-defined value. For beta decay, however, the observed broad distribution of energies suggested that energy is lost in the beta decay process. This spectrum was puzzling for many years.	1,194	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	904	14	2,024	8	10	31	2	0
A second problem is related to the conservation of angular momentum. Molecular band spectra showed that the nuclear spin of nitrogen-14 is 1 (i.e., equal to the reduced Planck constant) and more generally that the spin is integral for nuclei of even mass number and half-integral for nuclei of odd mass number. This was later explained by the proton-neutron model of the nucleus. Beta decay leaves the mass number unchanged, so the change of nuclear spin must be an integer. However, the electron spin is 1/2, hence angular momentum would not be conserved if beta decay were simply electron emission.	600	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	905	15	2,024	8	10	31	2	1	CONSTANT
From 1920 to 1927, Charles Drummond Ellis (along with Chadwick and colleagues) further established that the beta decay spectrum is continuous. In 1933, Ellis and Nevill Mott obtained strong evidence that the beta spectrum has an effective upper bound in energy. Niels Bohr had suggested that the beta spectrum could be explained if conservation of energy was true only in a statistical sense, thus this principle might be violated in any given decay. However, the upper bound in beta energies determined by Ellis and Mott ruled out that notion. Now, the problem of how to account for the variability of energy in known beta decay products, as well as for conservation of momentum and angular momentum in the process, became acute.	730	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	906	16	2,024	8	10	31	2	0
In a famous letter written in 1930, Wolfgang Pauli attempted to resolve the beta-particle energy conundrum by suggesting that, in addition to electrons and protons, atomic nuclei also contained an extremely light neutral particle, which he called the neutron. He suggested that this "neutron" was also emitted during beta decay (thus accounting for the known missing energy, momentum, and angular momentum), but it had simply not yet been observed. In 1931, Enrico Fermi renamed Pauli's "neutron" the "neutrino" ('little neutral one' in Italian). In 1933, Fermi published his landmark theory for beta decay, where he applied the principles of quantum mechanics to matter particles, supposing that they can be created and annihilated, just as the light quanta in atomic transitions. Thus, according to Fermi, neutrinos are created in the beta-decay process, rather than contained in the nucleus; the same happens to electrons. The neutrino interaction with matter was so weak that detecting it proved a severe experimental challenge. Further indirect evidence of the existence of the neutrino was obtained by observing the recoil of nuclei that emitted such a particle after absorbing an electron. Neutrinos were finally detected directly in 1956 by the American physicists Clyde Cowan and Frederick Reines in the Cowan–Reines neutrino experiment. The properties of neutrinos were (with a few minor modifications) as predicted by Pauli and Fermi.	1,445	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	907	17	2,024	8	10	31	2	0
In 1934, Frédéric and Irène Joliot-Curie bombarded aluminium with alpha particles to effect the nuclear reaction 2 He + 13 Al → 15 P + 0 n, and observed that the product isotope 15 P emits a positron identical to those found in cosmic rays (discovered by Carl David Anderson in 1932). This was the first example of β decay (positron emission), which they termed artificial radioactivity since 15 P is a short-lived nuclide which does not exist in nature. In recognition of their discovery, the couple were awarded the Nobel Prize in Chemistry in 1935.	551	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	908	18	2,024	8	10	31	2	0
The theory of electron capture was first discussed by Gian-Carlo Wick in a 1934 paper, and then developed by Hideki Yukawa and others. K-electron capture was first observed in 1937 by Luis Alvarez, in the nuclide V. Alvarez went on to study electron capture in Ga and other nuclides.	283	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	909	19	2,024	8	10	31	2	0
In 1956, Tsung-Dao Lee and Chen Ning Yang noticed that there was no evidence that parity was conserved in weak interactions, and so they postulated that this symmetry may not be preserved by the weak force. They sketched the design for an experiment for testing conservation of parity in the laboratory. Later that year, Chien-Shiung Wu and coworkers conducted the Wu experiment showing an asymmetrical beta decay of Co at cold temperatures that proved that parity is not conserved in beta decay. This surprising result overturned long-held assumptions about parity and the weak force. In recognition of their theoretical work, Lee and Yang were awarded the Nobel Prize for Physics in 1957. However Wu, who was female, was not awarded the Nobel prize.	751	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	910	20	2,024	8	10	31	2	0
In β decay, the weak interaction converts an atomic nucleus into a nucleus with atomic number increased by one, while emitting an electron (e) and an electron antineutrino (ν e). β decay generally occurs in neutron-rich nuclei. The generic equation is:	252	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	911	21	2,024	8	10	31	2	0
where A and Z are the mass number and atomic number of the decaying nucleus, and X and X′ are the initial and final elements, respectively.	139	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	912	22	2,024	8	10	31	2	0
Another example is when the free neutron (0 n) decays by β decay into a proton (p):	83	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	913	23	2,024	8	10	31	2	0
At the fundamental level (as depicted in the Feynman diagram on the right), this is caused by the conversion of the negatively charged (− ⁠ 1 / 3 ⁠ e) down quark to the positively charged (+ ⁠ 2 / 3 ⁠ e) up quark by emission of a W boson ; the W boson subsequently decays into an electron and an electron antineutrino:	318	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	914	24	2,024	8	10	31	2	0
In β decay, or positron emission, the weak interaction converts an atomic nucleus into a nucleus with atomic number decreased by one, while emitting a positron (e) and an electron neutrino (ν e). β decay generally occurs in proton-rich nuclei. The generic equation is:	268	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	915	25	2,024	8	10	31	2	0
This may be considered as the decay of a proton inside the nucleus to a neutron:	80	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	916	26	2,024	8	10	31	2	0
However, β decay cannot occur in an isolated proton because it requires energy, due to the mass of the neutron being greater than the mass of the proton. β decay can only happen inside nuclei when the daughter nucleus has a greater binding energy (and therefore a lower total energy) than the mother nucleus. The difference between these energies goes into the reaction of converting a proton into a neutron, a positron, and a neutrino and into the kinetic energy of these particles. This process is opposite to negative beta decay, in that the weak interaction converts a proton into a neutron by converting an up quark into a down quark resulting in the emission of a W or the absorption of a W. When a W boson is emitted, it decays into a positron and an electron neutrino :	777	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	917	27	2,024	8	10	31	2	0
In all cases where β decay (positron emission) of a nucleus is allowed energetically, so too is electron capture allowed. This is a process during which a nucleus captures one of its atomic electrons, resulting in the emission of a neutrino:	241	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	918	28	2,024	8	10	31	2	0
An example of electron capture is one of the decay modes of krypton-81 into bromine-81 :	88	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	919	29	2,024	8	10	31	2	0
All emitted neutrinos are of the same energy. In proton-rich nuclei where the energy difference between the initial and final states is less than 2 m e c, β decay is not energetically possible, and electron capture is the sole decay mode.	238	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	920	30	2,024	8	10	31	2	0
If the captured electron comes from the innermost shell of the atom, the K-shell, which has the highest probability to interact with the nucleus, the process is called K-capture. If it comes from the L-shell, the process is called L-capture, etc.	246	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	921	31	2,024	8	10	31	2	0
Electron capture is a competing (simultaneous) decay process for all nuclei that can undergo β decay. The converse, however, is not true: electron capture is the only type of decay that is allowed in proton-rich nuclides that do not have sufficient energy to emit a positron and neutrino.	288	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	922	32	2,024	8	10	31	2	0
If the proton and neutron are part of an atomic nucleus, the above described decay processes transmute one chemical element into another. For example:	150	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	923	33	2,024	8	10	31	2	0
Beta decay does not change the number (A) of nucleons in the nucleus, but changes only its charge Z. Thus the set of all nuclides with the same A can be introduced; these isobaric nuclides may turn into each other via beta decay. For a given A there is one that is most stable. It is said to be beta stable, because it presents a local minimum of the mass excess : if such a nucleus has (A, Z) numbers, the neighbour nuclei (A, Z −1) and (A, Z +1) have higher mass excess and can beta decay into (A, Z), but not vice versa. For all odd mass numbers A, there is only one known beta-stable isobar. For even A, there are up to three different beta-stable isobars experimentally known; for example, 50 Sn, 52 Te, and 54 Xe are all beta-stable. There are about 350 known beta-decay stable nuclides.	793	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	924	34	2,024	8	10	31	2	0
Usually unstable nuclides are clearly either "neutron rich" or "proton rich", with the former undergoing beta decay and the latter undergoing electron capture (or more rarely, due to the higher energy requirements, positron decay). However, in a few cases of odd-proton, odd-neutron radionuclides, it may be energetically favorable for the radionuclide to decay to an even-proton, even-neutron isobar either by undergoing beta-positive or beta-negative decay. An often-cited example is the single isotope 29 Cu (29 protons, 35 neutrons), which illustrates three types of beta decay in competition. Copper-64 has a half-life of about 12.7 hours. This isotope has one unpaired proton and one unpaired neutron, so either the proton or the neutron can decay. This particular nuclide (though not all nuclides in this situation) is almost equally likely to decay through proton decay by positron emission (18%) or electron capture (43%) to 28 Ni, as it is through neutron decay by electron emission (39%) to 30 Zn.	1,008	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	925	35	2,024	8	10	31	2	0
Most naturally occurring nuclides on earth are beta stable. Nuclides that are not beta stable have half-lives ranging from under a second to periods of time significantly greater than the age of the universe. One common example of a long-lived isotope is the odd-proton odd-neutron nuclide 19 K, which undergoes all three types of beta decay (β, β and electron capture) with a half-life of 1.277 × 10 years.	407	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	926	36	2,024	8	10	31	2	0
B = n q − − n q ¯ ¯ 3 where	27	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	927	37	2,024	8	10	31	2	0
Beta decay just changes neutron to proton or, in the case of positive beta decay (electron capture) proton to neutron so the number of individual quarks doesn't change. It is only the baryon flavor that changes, here labelled as the isospin.	241	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	928	38	2,024	8	10	31	2	0
Up and down quarks have total isospin I = 1 2 and isospin projections I z = { 1 2 up quark − − 1 2 down quark	109	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	929	39	2,024	8	10	31	2	0
All other quarks have I = 0.	28	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	930	40	2,024	8	10	31	2	0
In general I z = 1 2 ( n u − − n d )	36	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	931	41	2,024	8	10	31	2	0
L ≡ ≡ n ℓ ℓ − − n ℓ ℓ ¯ ¯	25	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	932	42	2,024	8	10	31	2	0
so all leptons have assigned a value of +1, antileptons −1, and non-leptonic particles 0. n → → p + e − − + ν ν ¯ ¯ e L : 0 = 0 + 1 − − 1	137	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	933	43	2,024	8	10	31	2	0
For allowed decays, the net orbital angular momentum is zero, hence only spin quantum numbers are considered.	109	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	934	44	2,024	8	10	31	2	0
The electron and antineutrino are fermions, spin-1/2 objects, therefore they may couple to total S = 1 (parallel) or S = 0 (anti-parallel).	139	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	935	45	2,024	8	10	31	2	0
For forbidden decays, orbital angular momentum must also be taken into consideration.	85	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	936	46	2,024	8	10	31	2	0
The Q value is defined as the total energy released in a given nuclear decay. In beta decay, Q is therefore also the sum of the kinetic energies of the emitted beta particle, neutrino, and recoiling nucleus. (Because of the large mass of the nucleus compared to that of the beta particle and neutrino, the kinetic energy of the recoiling nucleus can generally be neglected.) Beta particles can therefore be emitted with any kinetic energy ranging from 0 to Q. A typical Q is around 1 MeV, but can range from a few keV to a few tens of MeV.	539	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	937	47	2,024	8	10	31	2	0
Since the rest mass of the electron is 511 keV, the most energetic beta particles are ultrarelativistic, with speeds very close to the speed of light. In the case of Re, the maximum speed of the beta particle is only 9.8% of the speed of light.	244	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	938	48	2,024	8	10	31	2	0
The following table gives some examples:	40	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	939	49	2,024	8	10	31	2	0
Consider the generic equation for beta decay	44	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	940	50	2,024	8	10	31	2	0
The Q value for this decay is	29	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	941	51	2,024	8	10	31	2	0
where m N ( X Z A ) is the mass of the nucleus of the Z X atom, m e is the mass of the electron, and m ν ν ¯ ¯ e is the mass of the electron antineutrino. In other words, the total energy released is the mass energy of the initial nucleus, minus the mass energy of the final nucleus, electron, and antineutrino. The mass of the nucleus m N is related to the standard atomic mass m by m ( X Z A ) c 2 = m N ( X Z A ) c 2 + Z m e c 2 − − ∑ ∑ i = 1 Z B i . That is, the total atomic mass is the mass of the nucleus, plus the mass of the electrons, minus the sum of all electron binding energies B i for the atom. This equation is rearranged to find m N ( X Z A ) , and m N ( X Z + 1 A ′ ) is found similarly. Substituting these nuclear masses into the Q -value equation, while neglecting the nearly-zero antineutrino mass and the difference in electron binding energies, which is very small for high- Z atoms, we have Q = [ m ( X Z A ) − − m ( X Z + 1 A ′ ) ] c 2 This energy is carried away as kinetic energy by the electron and antineutrino.	1,040	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	942	52	2,024	8	10	31	2	0
Because the reaction will proceed only when the Q value is positive, β decay can occur when the mass of atom Z X is greater than the mass of atom Z +1 X′.	154	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	943	53	2,024	8	10	31	2	0
The equations for β decay are similar, with the generic equation	64	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	944	54	2,024	8	10	31	2	0
giving Q = [ m N ( X Z A ) − − m N ( X Z − − 1 A ′ ) − − m e − − m ν ν e ] c 2 . However, in this equation, the electron masses do not cancel, and we are left with Q = [ m ( X Z A ) − − m ( X Z − − 1 A ′ ) − − 2 m e ] c 2 .	223	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	945	55	2,024	8	10	31	2	0
Because the reaction will proceed only when the Q value is positive, β decay can occur when the mass of atom Z X exceeds that of Z -1 X′ by at least twice the mass of the electron.	180	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	946	56	2,024	8	10	31	2	0
The analogous calculation for electron capture must take into account the binding energy of the electrons. This is because the atom will be left in an excited state after capturing the electron, and the binding energy of the captured innermost electron is significant. Using the generic equation for electron capture	316	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	947	57	2,024	8	10	31	2	0
we have Q = [ m N ( X Z A ) + m e − − m N ( X Z − − 1 A ′ ) − − m ν ν e ] c 2 , which simplifies to Q = [ m ( X Z A ) − − m ( X Z − − 1 A ′ ) ] c 2 − − B n , where B n is the binding energy of the captured electron.	215	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	948	58	2,024	8	10	31	2	0
Because the binding energy of the electron is much less than the mass of the electron, nuclei that can undergo β decay can always also undergo electron capture, but the reverse is not true.	189	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	949	59	2,024	8	10	31	2	0
Beta decay can be considered as a perturbation as described in quantum mechanics, and thus Fermi's Golden Rule can be applied. This leads to an expression for the kinetic energy spectrum N (T) of emitted betas as follows:	221	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	950	60	2,024	8	10	31	2	0
N ( T ) = C L ( T ) F ( Z , T ) p E ( Q − − T ) 2	49	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	951	61	2,024	8	10	31	2	0
where T is the kinetic energy, C L is a shape function that depends on the forbiddenness of the decay (it is constant for allowed decays), F (Z, T) is the Fermi Function (see below) with Z the charge of the final-state nucleus, E = T + mc is the total energy, p = ( E / c ) 2 − − ( m c ) 2 is the momentum, and Q is the Q value of the decay. The kinetic energy of the emitted neutrino is given approximately by Q minus the kinetic energy of the beta.	450	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	952	62	2,024	8	10	31	2	0
As an example, the beta decay spectrum of Bi (originally called RaE) is shown to the right.	91	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	953	63	2,024	8	10	31	2	0
The Fermi function that appears in the beta spectrum formula accounts for the Coulomb attraction / repulsion between the emitted beta and the final state nucleus. Approximating the associated wavefunctions to be spherically symmetric, the Fermi function can be analytically calculated to be:	291	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	954	64	2,024	8	10	31	2	0
F ( Z , T ) = 2 ( 1 + S ) Γ Γ ( 1 + 2 S ) 2 ( 2 p ρ ρ ) 2 S − − 2 e π π η η \| Γ Γ ( S + i η η ) \| 2 ,	101	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	955	65	2,024	8	10	31	2	0
where p is the final momentum, Γ the Gamma function, and (if α is the fine-structure constant and r N the radius of the final state nucleus) S = 1 − − α α 2 Z 2 , η η = ± ± Z e 2 E / ( ℏ ℏ c p ) (+ for electrons, − for positrons), and ρ ρ = r N / ℏ ℏ .	252	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	956	66	2,024	8	10	31	2	0
For non-relativistic betas (Q ≪ m e c), this expression can be approximated by:	79	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	957	67	2,024	8	10	31	2	0
F ( Z , T ) ≈ ≈ 2 π π η η 1 − − e − − 2 π π η η .	49	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	958	68	2,024	8	10	31	2	0
Other approximations can be found in the literature.	52	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	959	69	2,024	8	10	31	2	0
A Kurie plot (also known as a Fermi–Kurie plot) is a graph used in studying beta decay developed by Franz N. D. Kurie, in which the square root of the number of beta particles whose momenta (or energy) lie within a certain narrow range, divided by the Fermi function, is plotted against beta-particle energy. It is a straight line for allowed transitions and some forbidden transitions, in accord with the Fermi beta-decay theory. The energy-axis (x-axis) intercept of a Kurie plot corresponds to the maximum energy imparted to the electron/positron (the decay's Q value). With a Kurie plot one can find the limit on the effective mass of a neutrino.	650	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	960	70	2,024	8	10	31	2	0
After the discovery of parity non-conservation (see History), it was found that, in beta decay, electrons are emitted mostly with negative helicity, i.e., they move, naively speaking, like left-handed screws driven into a material (they have negative longitudinal polarization). Conversely, positrons have mostly positive helicity, i.e., they move like right-handed screws. Neutrinos (emitted in positron decay) have negative helicity, while antineutrinos (emitted in electron decay) have positive helicity.	507	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	961	71	2,024	8	10	31	2	0
The higher the energy of the particles, the higher their polarization.	70	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	962	72	2,024	8	10	31	2	0
Beta decays can be classified according to the angular momentum (L value) and total spin (S value) of the emitted radiation. Since total angular momentum must be conserved, including orbital and spin angular momentum, beta decay occurs by a variety of quantum state transitions to various nuclear angular momentum or spin states, known as "Fermi" or "Gamow–Teller" transitions. When beta decay particles carry no angular momentum (L = 0), the decay is referred to as "allowed", otherwise it is "forbidden".	506	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	963	73	2,024	8	10	31	2	0
Other decay modes, which are rare, are known as bound state decay and double beta decay.	88	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	964	74	2,024	8	10	31	2	0
A Fermi transition is a beta decay in which the spins of the emitted electron (positron) and anti-neutrino (neutrino) couple to total spin S = 0 , leading to an angular momentum change Δ Δ J = 0 between the initial and final states of the nucleus (assuming an allowed transition). In the non-relativistic limit, the nuclear part of the operator for a Fermi transition is given by O F = G V ∑ ∑ a τ τ ^ ^ a ± ± with G V the weak vector coupling constant, τ τ ± ± the isospin raising and lowering operators, and a running over all protons and neutrons in the nucleus.	565	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	965	75	2,024	8	10	31	2	0
A Gamow–Teller transition is a beta decay in which the spins of the emitted electron (positron) and anti-neutrino (neutrino) couple to total spin S = 1 , leading to an angular momentum change Δ Δ J = 0 , ± ± 1 between the initial and final states of the nucleus (assuming an allowed transition). In this case, the nuclear part of the operator is given by O G T = G A ∑ ∑ a σ σ ^ ^ a τ τ ^ ^ a ± ± with G A the weak axial-vector coupling constant, and σ σ the spin Pauli matrices, which can produce a spin-flip in the decaying nucleon.	534	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	966	76	2,024	8	10	31	2	0
When L > 0, the decay is referred to as " forbidden ". Nuclear selection rules require high L values to be accompanied by changes in nuclear spin (J) and parity (π). The selection rules for the L th forbidden transitions are: Δ Δ J = L − − 1 , L , L + 1 ; Δ Δ π π = ( − − 1 ) L , where Δ π = 1 or −1 corresponds to no parity change or parity change, respectively. The special case of a transition between isobaric analogue states, where the structure of the final state is very similar to the structure of the initial state, is referred to as "superallowed" for beta decay, and proceeds very quickly. The following table lists the Δ J and Δ π values for the first few values of L :	681	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	967	77	2,024	8	10	31	2	0
A very small minority of free neutron decays (about four per million) are so-called "two-body decays", in which the proton, electron and antineutrino are produced, but the electron fails to gain the 13.6 eV energy necessary to escape the proton, and therefore simply remains bound to it, as a neutral hydrogen atom. In this type of beta decay, in essence all of the neutron decay energy is carried off by the antineutrino.	422	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	968	78	2,024	8	10	31	2	0
For fully ionized atoms (bare nuclei), it is possible in likewise manner for electrons to fail to escape the atom, and to be emitted from the nucleus into low-lying atomic bound states (orbitals). This cannot occur for neutral atoms with low-lying bound states which are already filled by electrons.	299	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	969	79	2,024	8	10	31	2	0
Bound-state β decays were predicted by Daudel, Jean, and Lecoin in 1947, and the phenomenon in fully ionized atoms was first observed for Dy in 1992 by Jung et al. of the Darmstadt Heavy-Ion Research Center. Although neutral Dy is a stable isotope, the fully ionized Dy undergoes β decay into the K and L shells with a half-life of 47 days. The resulting nucleus - Ho - is stable only in the fully ionized state and will decay via electron capture into Dy in the neutral state. The half life for neutral Ho is 4750 years.	521	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	970	80	2,024	8	10	31	2	0
Another possibility is that a fully ionized atom undergoes greatly accelerated β decay, as observed for Re by Bosch et al., also at Darmstadt. Neutral Re does undergo β decay with a half-life of 41.6 × 10 years, but for fully ionized Re this is shortened to only 32.9 years. For comparison the variation of decay rates of other nuclear processes due to chemical environment is less than 1%.	390	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	971	81	2,024	8	10	31	2	0
Some nuclei can undergo double beta decay (ββ decay) where the charge of the nucleus changes by two units. Double beta decay is difficult to study, as the process has an extremely long half-life. In nuclei for which both β decay and ββ decay are possible, the rarer ββ decay process is effectively impossible to observe. However, in nuclei where β decay is forbidden but ββ decay is allowed, the process can be seen and a half-life measured. Thus, ββ decay is usually studied only for beta stable nuclei. Like single beta decay, double beta decay does not change A ; thus, at least one of the nuclides with some given A has to be stable with regard to both single and double beta decay.	686	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	972	82	2,024	8	10	31	2	0
"Ordinary" double beta decay results in the emission of two electrons and two antineutrinos. If neutrinos are Majorana particles (i.e., they are their own antiparticles), then a decay known as neutrinoless double beta decay will occur. Most neutrino physicists believe that neutrinoless double beta decay has never been observed.	329	Beta_decay	https://en.wikipedia.org/wiki/Beta_decay	973	83	2,024	8	10	31	2	0
Isaac Newton 's rotating bucket argument (also known as Newton's bucket) was designed to demonstrate that true rotational motion cannot be defined as the relative rotation of the body with respect to the immediately surrounding bodies. It is one of five arguments from the "properties, causes, and effects" of "true motion and rest" that support his contention that, in general, true motion and rest cannot be defined as special instances of motion or rest relative to other bodies, but instead can be defined only by reference to absolute space. Alternatively, these experiments provide an operational definition of what is meant by " absolute rotation ", and do not pretend to address the question of "rotation relative to what?" General relativity dispenses with absolute space and with physics whose cause is external to the system, with the concept of geodesics of spacetime.	880	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	974	0	2,024	8	10	31	2	0
These arguments, and a discussion of the distinctions between absolute and relative time, space, place and motion, appear in a scholium at the end of Definitions sections in Book I of Newton's work, The Mathematical Principles of Natural Philosophy (1687) (not to be confused with General Scholium at the end of Book III), which established the foundations of classical mechanics and introduced his law of universal gravitation, which yielded the first quantitatively adequate dynamical explanation of planetary motion.	519	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	975	1	2,024	8	10	31	2	0
Despite their embrace of the principle of rectilinear inertia and the recognition of the kinematical relativity of apparent motion (which underlies whether the Ptolemaic or the Copernican system is correct), natural philosophers of the seventeenth century continued to consider true motion and rest as physically separate descriptors of an individual body. The dominant view Newton opposed was devised by René Descartes, and was supported (in part) by Gottfried Leibniz. It held that empty space is a metaphysical impossibility because space is nothing other than the extension of matter, or, in other words, that when one speaks of the space between things one is actually making reference to the relationship that exists between those things and not to some entity that stands between them. Concordant with the above understanding, any assertion about the motion of a body boils down to a description over time in which the body under consideration is at t 1 found in the vicinity of one group of "landmark" bodies and at some t 2 is found in the vicinity of some other "landmark" body or bodies.	1,098	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	976	2	2,024	8	10	31	2	0
Descartes recognized that there would be a real difference, however, between a situation in which a body with movable parts and originally at rest with respect to a surrounding ring was itself accelerated to a certain angular velocity with respect to the ring, and another situation in which the surrounding ring were given a contrary acceleration with respect to the central object. With sole regard to the central object and the surrounding ring, the motions would be indistinguishable from each other assuming that both the central object and the surrounding ring were absolutely rigid objects. However, if neither the central object nor the surrounding ring were absolutely rigid then the parts of one or both of them would tend to fly out from the axis of rotation.	770	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	977	3	2,024	8	10	31	2	0
For contingent reasons having to do with the Inquisition, Descartes spoke of motion as both absolute and relative.	114	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	978	4	2,024	8	10	31	2	0
By the late 19th century, the contention that all motion is relative was re-introduced, notably by Ernst Mach (1883).	117	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	979	5	2,024	8	10	31	2	0
When, accordingly, we say that a body preserves unchanged its direction and velocity in space, our assertion is nothing more or less than an abbreviated reference to the entire universe.	186	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	980	6	2,024	8	10	31	2	0
Newton discusses a bucket (Latin : situla) filled with water hung by a cord. If the cord is twisted up tightly on itself and then the bucket is released, it begins to spin rapidly, not only with respect to the experimenter, but also in relation to the water it contains. (This situation would correspond to diagram B above.)	324	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	981	7	2,024	8	10	31	2	0
Although the relative motion at this stage is the greatest, the surface of the water remains flat, indicating that the parts of the water have no tendency to recede from the axis of relative motion, despite proximity to the pail. Eventually, as the cord continues to unwind, the surface of the water assumes a concave shape as it acquires the motion of the bucket spinning relative to the experimenter. This concave shape shows that the water is rotating, despite the fact that the water is at rest relative to the pail. In other words, it is not the relative motion of the pail and water that causes concavity of the water, contrary to the idea that motions can only be relative, and that there is no absolute motion. (This situation would correspond to diagram D.) Possibly the concavity of the water shows rotation relative to something else : say absolute space? Newton says: "One can find out and measure the true and absolute circular motion of the water".	962	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	982	8	2,024	8	10	31	2	0
In the 1846 Andrew Motte translation of Newton's words:	55	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	983	9	2,024	8	10	31	2	0
If a vessel, hung by a long cord, is so often turned about that the cord is strongly twisted, then filled with water, and held at rest together with the water; after, by the sudden action of another force, it is whirled about in the contrary way, and while the cord is untwisting itself, the vessel continues for some time this motion; the surface of the water will at first be plain, as before the vessel began to move; but the vessel by gradually communicating its motion to the water, will make it begin sensibly to revolve, and recede by little and little, and ascend to the sides of the vessel, forming itself into a concave figure... This ascent of the water shows its endeavour to recede from the axis of its motion; and the true and absolute circular motion of the water, which is here directly contrary to the relative, discovers itself, and may be measured by this endeavour.... And therefore, this endeavour does not depend upon any translation of the water in respect to ambient bodies, nor can true circular motion be defined by such translation....; but relative motions...are altogether destitute of any real effect.... It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from the apparent; because the parts of that immovable space in which these motions are performed, do by no means come under the observations of our senses.	1,416	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	984	10	2,024	8	10	31	2	0
The argument that the motion is absolute, not relative, is incomplete, as it limits the participants relevant to the experiment to only the pail and the water, a limitation that has not been established. In fact, the concavity of the water clearly involves gravitational attraction, and by implication the Earth also is a participant. Here is a critique due to Mach arguing that only relative motion is established:	415	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	985	11	2,024	8	10	31	2	0
Newton's experiment with the rotating vessel of water simply informs us that the relative rotation of the water with respect to the sides of the vessel produces no noticeable centrifugal forces, but that such forces are produced by its relative rotations with respect to the mass of the earth and other celestial bodies.	320	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	986	12	2,024	8	10	31	2	0
The degree in which Mach's hypothesis is integrated in general relativity is discussed in the article Mach's principle ; it is generally held that general relativity is not entirely Machian.	190	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	987	13	2,024	8	10	31	2	0
All observers agree that the surface of rotating water is curved. However, the explanation of this curvature involves centrifugal force for all observers with the exception of a truly stationary observer, who finds the curvature is consistent with the rate of rotation of the water as they observe it, with no need for an additional centrifugal force. Thus, a stationary frame can be identified, and it is not necessary to ask "Stationary with respect to what?":	462	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	988	14	2,024	8	10	31	2	0
The original question, "relative to what frame of reference do the laws of motion hold?" is revealed to be wrongly posed. For the laws of motion essentially determine a class of reference frames, and (in principle) a procedure for constructing them.	249	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	989	15	2,024	8	10	31	2	0
A supplementary thought experiment with the same objective of determining the occurrence of absolute rotation also was proposed by Newton: the example of observing two identical spheres in rotation about their center of gravity and tied together by a string. Occurrence of tension in the string is indicative of absolute rotation; see Rotating spheres.	352	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	990	16	2,024	8	10	31	2	0
The historic interest of the rotating bucket experiment is its usefulness in suggesting one can detect absolute rotation by observation of the shape of the surface of the water. However, one might question just how rotation brings about this change. Below are two approaches to understanding the concavity of the surface of rotating water in a bucket.	351	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	991	17	2,024	8	10	31	2	0
The shape of the surface of a rotating liquid in a bucket can be determined using Newton's laws for the various forces on an element of the surface. For example, see Knudsen and Hjorth. The analysis begins with the free body diagram in the co-rotating frame where the water appears stationary. The height of the water h = h (r) is a function of the radial distance r from the axis of rotation Ω, and the aim is to determine this function. An element of water volume on the surface is shown to be subject to three forces: the vertical force due to gravity F g, the horizontal, radially outward centrifugal force F Cfgl, and the force normal to the surface of the water F n due to the rest of the water surrounding the selected element of surface. The force due to surrounding water is known to be normal to the surface of the water because a liquid in equilibrium cannot support shear stresses. To quote Anthony and Brackett:	924	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	992	18	2,024	8	10	31	2	0
The surface of a fluid of uniform density..., if at rest, is everywhere perpendicular to the lines of force; for if this were not so, the force at a point on the surface could be resolved into two components, one perpendicular and the other tangent to the surface. But from the nature of a fluid, the tangential force would set up a motion of the fluid, which is contrary to the statement that the fluid is at rest.	415	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	993	19	2,024	8	10	31	2	0
Moreover, because the element of water does not move, the sum of all three forces must be zero. To sum to zero, the force of the water must point oppositely to the sum of the centrifugal and gravity forces, which means the surface of the water must adjust so its normal points in this direction. (A very similar problem is the design of a banked turn, where the slope of the turn is set so a car will not slide off the road. The analogy in the case of rotating bucket is that the element of water surface will "slide" up or down the surface unless the normal to the surface aligns with the vector resultant formed by the vector addition F g + F Cfgl.)	651	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	994	20	2,024	8	10	31	2	0
As r increases, the centrifugal force increases according to the relation (the equations are written per unit mass):	116	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	995	21	2,024	8	10	31	2	0
where Ω is the constant rate of rotation of the water. The gravitational force is unchanged at	94	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	996	22	2,024	8	10	31	2	0
where g is the acceleration due to gravity. These two forces add to make a resultant at an angle φ from the vertical given by	125	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	997	23	2,024	8	10	31	2	0
which clearly becomes larger as r increases. To ensure that this resultant is normal to the surface of the water, and therefore can be effectively nulled by the force of the water beneath, the normal to the surface must have the same angle, that is,	249	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	998	24	2,024	8	10	31	2	0
leading to the ordinary differential equation for the shape of the surface:	75	Bucket_argument	https://en.wikipedia.org/wiki/Bucket_argument	999	25	2,024	8	10	31	2	0

The beta spectrum, or distribution of energy values for the beta particles, is continuous. The total energy of the decay process is divided between the electron, the antineutrino, and the recoiling nuclide. In the figure to the right, an example of an electron with 0.40 MeV energy from the beta decay of Bi is shown. In this example, the total decay energy is 1.16 MeV, so the antineutrino has the remaining energy: 1.16 MeV − 0.40 MeV = 0.76 MeV. An electron at the far right of the curve would have the maximum possible kinetic energy, leaving the energy of the neutrino to be only its small rest mass.

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Radioactivity was discovered in 1896 by Henri Becquerel in uranium, and subsequently observed by Marie and Pierre Curie in thorium and in the new elements polonium and radium. In 1899, Ernest Rutherford separated radioactive emissions into two types: alpha and beta (now beta minus), based on penetration of objects and ability to cause ionization. Alpha rays could be stopped by thin sheets of paper or aluminium, whereas beta rays could penetrate several millimetres of aluminium. In 1900, Paul Villard identified a still more penetrating type of radiation, which Rutherford identified as a fundamentally new type in 1903 and termed gamma rays. Alpha, beta, and gamma are the first three letters of the Greek alphabet.

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In 1900, Becquerel measured the mass-to-charge ratio (m / e) for beta particles by the method of J.J. Thomson used to study cathode rays and identify the electron. He found that m / e for a beta particle is the same as for Thomson's electron, and therefore suggested that the beta particle is in fact an electron.

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In 1901, Rutherford and Frederick Soddy showed that alpha and beta radioactivity involves the transmutation of atoms into atoms of other chemical elements. In 1913, after the products of more radioactive decays were known, Soddy and Kazimierz Fajans independently proposed their radioactive displacement law, which states that beta (i.e., β) emission from one element produces another element one place to the right in the periodic table, while alpha emission produces an element two places to the left.

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The study of beta decay provided the first physical evidence for the existence of the neutrino. In both alpha and gamma decay, the resulting alpha or gamma particle has a narrow energy distribution, since the particle carries the energy from the difference between the initial and final nuclear states. However, the kinetic energy distribution, or spectrum, of beta particles measured by Lise Meitner and Otto Hahn in 1911 and by Jean Danysz in 1913 showed multiple lines on a diffuse background. These measurements offered the first hint that beta particles have a continuous spectrum. In 1914, James Chadwick used a magnetic spectrometer with one of Hans Geiger's new counters to make more accurate measurements which showed that the spectrum was continuous. The distribution of beta particle energies was in apparent contradiction to the law of conservation of energy. If beta decay were simply electron emission as assumed at the time, then the energy of the emitted electron should have a particular, well-defined value. For beta decay, however, the observed broad distribution of energies suggested that energy is lost in the beta decay process. This spectrum was puzzling for many years.

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A second problem is related to the conservation of angular momentum. Molecular band spectra showed that the nuclear spin of nitrogen-14 is 1 (i.e., equal to the reduced Planck constant) and more generally that the spin is integral for nuclei of even mass number and half-integral for nuclei of odd mass number. This was later explained by the proton-neutron model of the nucleus. Beta decay leaves the mass number unchanged, so the change of nuclear spin must be an integer. However, the electron spin is 1/2, hence angular momentum would not be conserved if beta decay were simply electron emission.

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CONSTANT

From 1920 to 1927, Charles Drummond Ellis (along with Chadwick and colleagues) further established that the beta decay spectrum is continuous. In 1933, Ellis and Nevill Mott obtained strong evidence that the beta spectrum has an effective upper bound in energy. Niels Bohr had suggested that the beta spectrum could be explained if conservation of energy was true only in a statistical sense, thus this principle might be violated in any given decay. However, the upper bound in beta energies determined by Ellis and Mott ruled out that notion. Now, the problem of how to account for the variability of energy in known beta decay products, as well as for conservation of momentum and angular momentum in the process, became acute.

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In a famous letter written in 1930, Wolfgang Pauli attempted to resolve the beta-particle energy conundrum by suggesting that, in addition to electrons and protons, atomic nuclei also contained an extremely light neutral particle, which he called the neutron. He suggested that this "neutron" was also emitted during beta decay (thus accounting for the known missing energy, momentum, and angular momentum), but it had simply not yet been observed. In 1931, Enrico Fermi renamed Pauli's "neutron" the "neutrino" ('little neutral one' in Italian). In 1933, Fermi published his landmark theory for beta decay, where he applied the principles of quantum mechanics to matter particles, supposing that they can be created and annihilated, just as the light quanta in atomic transitions. Thus, according to Fermi, neutrinos are created in the beta-decay process, rather than contained in the nucleus; the same happens to electrons. The neutrino interaction with matter was so weak that detecting it proved a severe experimental challenge. Further indirect evidence of the existence of the neutrino was obtained by observing the recoil of nuclei that emitted such a particle after absorbing an electron. Neutrinos were finally detected directly in 1956 by the American physicists Clyde Cowan and Frederick Reines in the Cowan–Reines neutrino experiment. The properties of neutrinos were (with a few minor modifications) as predicted by Pauli and Fermi.

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In 1934, Frédéric and Irène Joliot-Curie bombarded aluminium with alpha particles to effect the nuclear reaction 2 He + 13 Al → 15 P + 0 n, and observed that the product isotope 15 P emits a positron identical to those found in cosmic rays (discovered by Carl David Anderson in 1932). This was the first example of β decay (positron emission), which they termed artificial radioactivity since 15 P is a short-lived nuclide which does not exist in nature. In recognition of their discovery, the couple were awarded the Nobel Prize in Chemistry in 1935.

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The theory of electron capture was first discussed by Gian-Carlo Wick in a 1934 paper, and then developed by Hideki Yukawa and others. K-electron capture was first observed in 1937 by Luis Alvarez, in the nuclide V. Alvarez went on to study electron capture in Ga and other nuclides.

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In 1956, Tsung-Dao Lee and Chen Ning Yang noticed that there was no evidence that parity was conserved in weak interactions, and so they postulated that this symmetry may not be preserved by the weak force. They sketched the design for an experiment for testing conservation of parity in the laboratory. Later that year, Chien-Shiung Wu and coworkers conducted the Wu experiment showing an asymmetrical beta decay of Co at cold temperatures that proved that parity is not conserved in beta decay. This surprising result overturned long-held assumptions about parity and the weak force. In recognition of their theoretical work, Lee and Yang were awarded the Nobel Prize for Physics in 1957. However Wu, who was female, was not awarded the Nobel prize.

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In β decay, the weak interaction converts an atomic nucleus into a nucleus with atomic number increased by one, while emitting an electron (e) and an electron antineutrino (ν e). β decay generally occurs in neutron-rich nuclei. The generic equation is:

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where A and Z are the mass number and atomic number of the decaying nucleus, and X and X′ are the initial and final elements, respectively.

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Another example is when the free neutron (0 n) decays by β decay into a proton (p):

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At the fundamental level (as depicted in the Feynman diagram on the right), this is caused by the conversion of the negatively charged (− ⁠ 1 / 3 ⁠ e) down quark to the positively charged (+ ⁠ 2 / 3 ⁠ e) up quark by emission of a W boson ; the W boson subsequently decays into an electron and an electron antineutrino:

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In β decay, or positron emission, the weak interaction converts an atomic nucleus into a nucleus with atomic number decreased by one, while emitting a positron (e) and an electron neutrino (ν e). β decay generally occurs in proton-rich nuclei. The generic equation is:

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This may be considered as the decay of a proton inside the nucleus to a neutron:

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However, β decay cannot occur in an isolated proton because it requires energy, due to the mass of the neutron being greater than the mass of the proton. β decay can only happen inside nuclei when the daughter nucleus has a greater binding energy (and therefore a lower total energy) than the mother nucleus. The difference between these energies goes into the reaction of converting a proton into a neutron, a positron, and a neutrino and into the kinetic energy of these particles. This process is opposite to negative beta decay, in that the weak interaction converts a proton into a neutron by converting an up quark into a down quark resulting in the emission of a W or the absorption of a W. When a W boson is emitted, it decays into a positron and an electron neutrino :

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In all cases where β decay (positron emission) of a nucleus is allowed energetically, so too is electron capture allowed. This is a process during which a nucleus captures one of its atomic electrons, resulting in the emission of a neutrino:

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An example of electron capture is one of the decay modes of krypton-81 into bromine-81 :

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All emitted neutrinos are of the same energy. In proton-rich nuclei where the energy difference between the initial and final states is less than 2 m e c, β decay is not energetically possible, and electron capture is the sole decay mode.

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If the captured electron comes from the innermost shell of the atom, the K-shell, which has the highest probability to interact with the nucleus, the process is called K-capture. If it comes from the L-shell, the process is called L-capture, etc.

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2,024

8

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Electron capture is a competing (simultaneous) decay process for all nuclei that can undergo β decay. The converse, however, is not true: electron capture is the only type of decay that is allowed in proton-rich nuclides that do not have sufficient energy to emit a positron and neutrino.

288

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

922

32

2,024

8

10

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0

If the proton and neutron are part of an atomic nucleus, the above described decay processes transmute one chemical element into another. For example:

150

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

923

33

2,024

8

10

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2

0

Beta decay does not change the number (A) of nucleons in the nucleus, but changes only its charge Z. Thus the set of all nuclides with the same A can be introduced; these isobaric nuclides may turn into each other via beta decay. For a given A there is one that is most stable. It is said to be beta stable, because it presents a local minimum of the mass excess : if such a nucleus has (A, Z) numbers, the neighbour nuclei (A, Z −1) and (A, Z +1) have higher mass excess and can beta decay into (A, Z), but not vice versa. For all odd mass numbers A, there is only one known beta-stable isobar. For even A, there are up to three different beta-stable isobars experimentally known; for example, 50 Sn, 52 Te, and 54 Xe are all beta-stable. There are about 350 known beta-decay stable nuclides.

793

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

924

34

2,024

8

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2

0

Usually unstable nuclides are clearly either "neutron rich" or "proton rich", with the former undergoing beta decay and the latter undergoing electron capture (or more rarely, due to the higher energy requirements, positron decay). However, in a few cases of odd-proton, odd-neutron radionuclides, it may be energetically favorable for the radionuclide to decay to an even-proton, even-neutron isobar either by undergoing beta-positive or beta-negative decay. An often-cited example is the single isotope 29 Cu (29 protons, 35 neutrons), which illustrates three types of beta decay in competition. Copper-64 has a half-life of about 12.7 hours. This isotope has one unpaired proton and one unpaired neutron, so either the proton or the neutron can decay. This particular nuclide (though not all nuclides in this situation) is almost equally likely to decay through proton decay by positron emission (18%) or electron capture (43%) to 28 Ni, as it is through neutron decay by electron emission (39%) to 30 Zn.

1,008

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

925

35

2,024

8

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0

Most naturally occurring nuclides on earth are beta stable. Nuclides that are not beta stable have half-lives ranging from under a second to periods of time significantly greater than the age of the universe. One common example of a long-lived isotope is the odd-proton odd-neutron nuclide 19 K, which undergoes all three types of beta decay (β, β and electron capture) with a half-life of 1.277 × 10 years.

407

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

926

36

2,024

8

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2

0

B = n q − − n q ¯ ¯ 3 where

27

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

927

37

2,024

8

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0

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2

0

Beta decay just changes neutron to proton or, in the case of positive beta decay (electron capture) proton to neutron so the number of individual quarks doesn't change. It is only the baryon flavor that changes, here labelled as the isospin.

241

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

928

38

2,024

8

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0

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2

0

Up and down quarks have total isospin I = 1 2 and isospin projections I z = { 1 2 up quark − − 1 2 down quark

109

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

929

39

2,024

8

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0

31

2

0

All other quarks have I = 0.

28

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

930

40

2,024

8

10

0

31

2

0

In general I z = 1 2 ( n u − − n d )

36

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

931

41

2,024

8

10

0

31

2

0

L ≡ ≡ n ℓ ℓ − − n ℓ ℓ ¯ ¯

25

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

932

42

2,024

8

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0

so all leptons have assigned a value of +1, antileptons −1, and non-leptonic particles 0. n → → p + e − − + ν ν ¯ ¯ e L : 0 = 0 + 1 − − 1

137

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

933

43

2,024

8

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0

31

2

0

For allowed decays, the net orbital angular momentum is zero, hence only spin quantum numbers are considered.

109

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

934

44

2,024

8

10

0

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0

The electron and antineutrino are fermions, spin-1/2 objects, therefore they may couple to total S = 1 (parallel) or S = 0 (anti-parallel).

139

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

935

45

2,024

8

10

0

31

2

0

For forbidden decays, orbital angular momentum must also be taken into consideration.

85

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

936

46

2,024

8

10

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31

2

0

The Q value is defined as the total energy released in a given nuclear decay. In beta decay, Q is therefore also the sum of the kinetic energies of the emitted beta particle, neutrino, and recoiling nucleus. (Because of the large mass of the nucleus compared to that of the beta particle and neutrino, the kinetic energy of the recoiling nucleus can generally be neglected.) Beta particles can therefore be emitted with any kinetic energy ranging from 0 to Q. A typical Q is around 1 MeV, but can range from a few keV to a few tens of MeV.

539

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

937

47

2,024

8

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2

0

Since the rest mass of the electron is 511 keV, the most energetic beta particles are ultrarelativistic, with speeds very close to the speed of light. In the case of Re, the maximum speed of the beta particle is only 9.8% of the speed of light.

244

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

938

48

2,024

8

10

0

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2

0

The following table gives some examples:

40

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

939

49

2,024

8

10

0

31

2

0

Consider the generic equation for beta decay

44

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

940

50

2,024

8

10

0

31

2

0

The Q value for this decay is

29

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

941

51

2,024

8

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2

0

where m N ( X Z A ) is the mass of the nucleus of the Z X atom, m e is the mass of the electron, and m ν ν ¯ ¯ e is the mass of the electron antineutrino. In other words, the total energy released is the mass energy of the initial nucleus, minus the mass energy of the final nucleus, electron, and antineutrino. The mass of the nucleus m N is related to the standard atomic mass m by m ( X Z A ) c 2 = m N ( X Z A ) c 2 + Z m e c 2 − − ∑ ∑ i = 1 Z B i . That is, the total atomic mass is the mass of the nucleus, plus the mass of the electrons, minus the sum of all electron binding energies B i for the atom. This equation is rearranged to find m N ( X Z A ) , and m N ( X Z + 1 A ′ ) is found similarly. Substituting these nuclear masses into the Q -value equation, while neglecting the nearly-zero antineutrino mass and the difference in electron binding energies, which is very small for high- Z atoms, we have Q = [ m ( X Z A ) − − m ( X Z + 1 A ′ ) ] c 2 This energy is carried away as kinetic energy by the electron and antineutrino.

1,040

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

942

52

2,024

8

10

0

31

2

0

Because the reaction will proceed only when the Q value is positive, β decay can occur when the mass of atom Z X is greater than the mass of atom Z +1 X′.

154

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

943

53

2,024

8

10

0

31

2

0

The equations for β decay are similar, with the generic equation

64

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

944

54

2,024

8

10

0

31

2

0

giving Q = [ m N ( X Z A ) − − m N ( X Z − − 1 A ′ ) − − m e − − m ν ν e ] c 2 . However, in this equation, the electron masses do not cancel, and we are left with Q = [ m ( X Z A ) − − m ( X Z − − 1 A ′ ) − − 2 m e ] c 2 .

223

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

945

55

2,024

8

10

0

31

2

0

Because the reaction will proceed only when the Q value is positive, β decay can occur when the mass of atom Z X exceeds that of Z -1 X′ by at least twice the mass of the electron.

180

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

946

56

2,024

8

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0

31

2

0

The analogous calculation for electron capture must take into account the binding energy of the electrons. This is because the atom will be left in an excited state after capturing the electron, and the binding energy of the captured innermost electron is significant. Using the generic equation for electron capture

316

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

947

57

2,024

8

10

0

31

2

0

we have Q = [ m N ( X Z A ) + m e − − m N ( X Z − − 1 A ′ ) − − m ν ν e ] c 2 , which simplifies to Q = [ m ( X Z A ) − − m ( X Z − − 1 A ′ ) ] c 2 − − B n , where B n is the binding energy of the captured electron.

215

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

948

58

2,024

8

10

0

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2

0

Because the binding energy of the electron is much less than the mass of the electron, nuclei that can undergo β decay can always also undergo electron capture, but the reverse is not true.

189

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

949

59

2,024

8

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0

Beta decay can be considered as a perturbation as described in quantum mechanics, and thus Fermi's Golden Rule can be applied. This leads to an expression for the kinetic energy spectrum N (T) of emitted betas as follows:

221

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

950

60

2,024

8

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0

31

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0

N ( T ) = C L ( T ) F ( Z , T ) p E ( Q − − T ) 2

49

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

951

61

2,024

8

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2

0

where T is the kinetic energy, C L is a shape function that depends on the forbiddenness of the decay (it is constant for allowed decays), F (Z, T) is the Fermi Function (see below) with Z the charge of the final-state nucleus, E = T + mc is the total energy, p = ( E / c ) 2 − − ( m c ) 2 is the momentum, and Q is the Q value of the decay. The kinetic energy of the emitted neutrino is given approximately by Q minus the kinetic energy of the beta.

450

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

952

62

2,024

8

10

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0

As an example, the beta decay spectrum of Bi (originally called RaE) is shown to the right.

91

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

953

63

2,024

8

10

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2

0

The Fermi function that appears in the beta spectrum formula accounts for the Coulomb attraction / repulsion between the emitted beta and the final state nucleus. Approximating the associated wavefunctions to be spherically symmetric, the Fermi function can be analytically calculated to be:

291

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

954

64

2,024

8

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0

F ( Z , T ) = 2 ( 1 + S ) Γ Γ ( 1 + 2 S ) 2 ( 2 p ρ ρ ) 2 S − − 2 e π π η η | Γ Γ ( S + i η η ) | 2 ,

101

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

955

65

2,024

8

10

0

31

2

0

where p is the final momentum, Γ the Gamma function, and (if α is the fine-structure constant and r N the radius of the final state nucleus) S = 1 − − α α 2 Z 2 , η η = ± ± Z e 2 E / ( ℏ ℏ c p ) (+ for electrons, − for positrons), and ρ ρ = r N / ℏ ℏ .

252

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

956

66

2,024

8

10

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2

0

For non-relativistic betas (Q ≪ m e c), this expression can be approximated by:

79

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

957

67

2,024

8

10

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0

F ( Z , T ) ≈ ≈ 2 π π η η 1 − − e − − 2 π π η η .

49

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

958

68

2,024

8

10

0

31

2

0

Other approximations can be found in the literature.

52

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

959

69

2,024

8

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0

A Kurie plot (also known as a Fermi–Kurie plot) is a graph used in studying beta decay developed by Franz N. D. Kurie, in which the square root of the number of beta particles whose momenta (or energy) lie within a certain narrow range, divided by the Fermi function, is plotted against beta-particle energy. It is a straight line for allowed transitions and some forbidden transitions, in accord with the Fermi beta-decay theory. The energy-axis (x-axis) intercept of a Kurie plot corresponds to the maximum energy imparted to the electron/positron (the decay's Q value). With a Kurie plot one can find the limit on the effective mass of a neutrino.

650

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

960

70

2,024

8

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2

0

After the discovery of parity non-conservation (see History), it was found that, in beta decay, electrons are emitted mostly with negative helicity, i.e., they move, naively speaking, like left-handed screws driven into a material (they have negative longitudinal polarization). Conversely, positrons have mostly positive helicity, i.e., they move like right-handed screws. Neutrinos (emitted in positron decay) have negative helicity, while antineutrinos (emitted in electron decay) have positive helicity.

507

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

961

71

2,024

8

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0

The higher the energy of the particles, the higher their polarization.

70

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

962

72

2,024

8

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0

Beta decays can be classified according to the angular momentum (L value) and total spin (S value) of the emitted radiation. Since total angular momentum must be conserved, including orbital and spin angular momentum, beta decay occurs by a variety of quantum state transitions to various nuclear angular momentum or spin states, known as "Fermi" or "Gamow–Teller" transitions. When beta decay particles carry no angular momentum (L = 0), the decay is referred to as "allowed", otherwise it is "forbidden".

506

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

963

73

2,024

8

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0

Other decay modes, which are rare, are known as bound state decay and double beta decay.

88

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

964

74

2,024

8

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A Fermi transition is a beta decay in which the spins of the emitted electron (positron) and anti-neutrino (neutrino) couple to total spin S = 0 , leading to an angular momentum change Δ Δ J = 0 between the initial and final states of the nucleus (assuming an allowed transition). In the non-relativistic limit, the nuclear part of the operator for a Fermi transition is given by O F = G V ∑ ∑ a τ τ ^ ^ a ± ± with G V the weak vector coupling constant, τ τ ± ± the isospin raising and lowering operators, and a running over all protons and neutrons in the nucleus.

565

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

965

75

2,024

8

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0

A Gamow–Teller transition is a beta decay in which the spins of the emitted electron (positron) and anti-neutrino (neutrino) couple to total spin S = 1 , leading to an angular momentum change Δ Δ J = 0 , ± ± 1 between the initial and final states of the nucleus (assuming an allowed transition). In this case, the nuclear part of the operator is given by O G T = G A ∑ ∑ a σ σ ^ ^ a τ τ ^ ^ a ± ± with G A the weak axial-vector coupling constant, and σ σ the spin Pauli matrices, which can produce a spin-flip in the decaying nucleon.

534

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

966

76

2,024

8

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0

When L > 0, the decay is referred to as " forbidden ". Nuclear selection rules require high L values to be accompanied by changes in nuclear spin (J) and parity (π). The selection rules for the L th forbidden transitions are: Δ Δ J = L − − 1 , L , L + 1 ; Δ Δ π π = ( − − 1 ) L , where Δ π = 1 or −1 corresponds to no parity change or parity change, respectively. The special case of a transition between isobaric analogue states, where the structure of the final state is very similar to the structure of the initial state, is referred to as "superallowed" for beta decay, and proceeds very quickly. The following table lists the Δ J and Δ π values for the first few values of L :

681

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

967

77

2,024

8

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A very small minority of free neutron decays (about four per million) are so-called "two-body decays", in which the proton, electron and antineutrino are produced, but the electron fails to gain the 13.6 eV energy necessary to escape the proton, and therefore simply remains bound to it, as a neutral hydrogen atom. In this type of beta decay, in essence all of the neutron decay energy is carried off by the antineutrino.

422

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

968

78

2,024

8

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0

For fully ionized atoms (bare nuclei), it is possible in likewise manner for electrons to fail to escape the atom, and to be emitted from the nucleus into low-lying atomic bound states (orbitals). This cannot occur for neutral atoms with low-lying bound states which are already filled by electrons.

299

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

969

79

2,024

8

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0

Bound-state β decays were predicted by Daudel, Jean, and Lecoin in 1947, and the phenomenon in fully ionized atoms was first observed for Dy in 1992 by Jung et al. of the Darmstadt Heavy-Ion Research Center. Although neutral Dy is a stable isotope, the fully ionized Dy undergoes β decay into the K and L shells with a half-life of 47 days. The resulting nucleus - Ho - is stable only in the fully ionized state and will decay via electron capture into Dy in the neutral state. The half life for neutral Ho is 4750 years.

521

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

970

80

2,024

8

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0

Another possibility is that a fully ionized atom undergoes greatly accelerated β decay, as observed for Re by Bosch et al., also at Darmstadt. Neutral Re does undergo β decay with a half-life of 41.6 × 10 years, but for fully ionized Re this is shortened to only 32.9 years. For comparison the variation of decay rates of other nuclear processes due to chemical environment is less than 1%.

390

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

971

81

2,024

8

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0

Some nuclei can undergo double beta decay (ββ decay) where the charge of the nucleus changes by two units. Double beta decay is difficult to study, as the process has an extremely long half-life. In nuclei for which both β decay and ββ decay are possible, the rarer ββ decay process is effectively impossible to observe. However, in nuclei where β decay is forbidden but ββ decay is allowed, the process can be seen and a half-life measured. Thus, ββ decay is usually studied only for beta stable nuclei. Like single beta decay, double beta decay does not change A ; thus, at least one of the nuclides with some given A has to be stable with regard to both single and double beta decay.

686

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

972

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8

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"Ordinary" double beta decay results in the emission of two electrons and two antineutrinos. If neutrinos are Majorana particles (i.e., they are their own antiparticles), then a decay known as neutrinoless double beta decay will occur. Most neutrino physicists believe that neutrinoless double beta decay has never been observed.

329

Beta_decay

https://en.wikipedia.org/wiki/Beta_decay

973

83

2,024

8

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2

0

Isaac Newton 's rotating bucket argument (also known as Newton's bucket) was designed to demonstrate that true rotational motion cannot be defined as the relative rotation of the body with respect to the immediately surrounding bodies. It is one of five arguments from the "properties, causes, and effects" of "true motion and rest" that support his contention that, in general, true motion and rest cannot be defined as special instances of motion or rest relative to other bodies, but instead can be defined only by reference to absolute space. Alternatively, these experiments provide an operational definition of what is meant by " absolute rotation ", and do not pretend to address the question of "rotation relative to what?" General relativity dispenses with absolute space and with physics whose cause is external to the system, with the concept of geodesics of spacetime.

880

Bucket_argument

https://en.wikipedia.org/wiki/Bucket_argument

974

0

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8

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0

These arguments, and a discussion of the distinctions between absolute and relative time, space, place and motion, appear in a scholium at the end of Definitions sections in Book I of Newton's work, The Mathematical Principles of Natural Philosophy (1687) (not to be confused with General Scholium at the end of Book III), which established the foundations of classical mechanics and introduced his law of universal gravitation, which yielded the first quantitatively adequate dynamical explanation of planetary motion.

519

Bucket_argument

https://en.wikipedia.org/wiki/Bucket_argument

975

1

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Despite their embrace of the principle of rectilinear inertia and the recognition of the kinematical relativity of apparent motion (which underlies whether the Ptolemaic or the Copernican system is correct), natural philosophers of the seventeenth century continued to consider true motion and rest as physically separate descriptors of an individual body. The dominant view Newton opposed was devised by René Descartes, and was supported (in part) by Gottfried Leibniz. It held that empty space is a metaphysical impossibility because space is nothing other than the extension of matter, or, in other words, that when one speaks of the space between things one is actually making reference to the relationship that exists between those things and not to some entity that stands between them. Concordant with the above understanding, any assertion about the motion of a body boils down to a description over time in which the body under consideration is at t 1 found in the vicinity of one group of "landmark" bodies and at some t 2 is found in the vicinity of some other "landmark" body or bodies.

1,098

Bucket_argument

https://en.wikipedia.org/wiki/Bucket_argument

976

2

2,024

8

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Descartes recognized that there would be a real difference, however, between a situation in which a body with movable parts and originally at rest with respect to a surrounding ring was itself accelerated to a certain angular velocity with respect to the ring, and another situation in which the surrounding ring were given a contrary acceleration with respect to the central object. With sole regard to the central object and the surrounding ring, the motions would be indistinguishable from each other assuming that both the central object and the surrounding ring were absolutely rigid objects. However, if neither the central object nor the surrounding ring were absolutely rigid then the parts of one or both of them would tend to fly out from the axis of rotation.

770

Bucket_argument

https://en.wikipedia.org/wiki/Bucket_argument

977

3

2,024

8

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0

For contingent reasons having to do with the Inquisition, Descartes spoke of motion as both absolute and relative.

114

Bucket_argument

https://en.wikipedia.org/wiki/Bucket_argument

978

4

2,024

8

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31

2

0

By the late 19th century, the contention that all motion is relative was re-introduced, notably by Ernst Mach (1883).

117

Bucket_argument

https://en.wikipedia.org/wiki/Bucket_argument

979

5

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When, accordingly, we say that a body preserves unchanged its direction and velocity in space, our assertion is nothing more or less than an abbreviated reference to the entire universe.

186

Bucket_argument

https://en.wikipedia.org/wiki/Bucket_argument

980

6

2,024

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Newton discusses a bucket (Latin : situla) filled with water hung by a cord. If the cord is twisted up tightly on itself and then the bucket is released, it begins to spin rapidly, not only with respect to the experimenter, but also in relation to the water it contains. (This situation would correspond to diagram B above.)

324

Bucket_argument

https://en.wikipedia.org/wiki/Bucket_argument

981

7

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Although the relative motion at this stage is the greatest, the surface of the water remains flat, indicating that the parts of the water have no tendency to recede from the axis of relative motion, despite proximity to the pail. Eventually, as the cord continues to unwind, the surface of the water assumes a concave shape as it acquires the motion of the bucket spinning relative to the experimenter. This concave shape shows that the water is rotating, despite the fact that the water is at rest relative to the pail. In other words, it is not the relative motion of the pail and water that causes concavity of the water, contrary to the idea that motions can only be relative, and that there is no absolute motion. (This situation would correspond to diagram D.) Possibly the concavity of the water shows rotation relative to something else : say absolute space? Newton says: "One can find out and measure the true and absolute circular motion of the water".

962

Bucket_argument

https://en.wikipedia.org/wiki/Bucket_argument

982

8

2,024

8

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In the 1846 Andrew Motte translation of Newton's words:

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If a vessel, hung by a long cord, is so often turned about that the cord is strongly twisted, then filled with water, and held at rest together with the water; after, by the sudden action of another force, it is whirled about in the contrary way, and while the cord is untwisting itself, the vessel continues for some time this motion; the surface of the water will at first be plain, as before the vessel began to move; but the vessel by gradually communicating its motion to the water, will make it begin sensibly to revolve, and recede by little and little, and ascend to the sides of the vessel, forming itself into a concave figure... This ascent of the water shows its endeavour to recede from the axis of its motion; and the true and absolute circular motion of the water, which is here directly contrary to the relative, discovers itself, and may be measured by this endeavour.... And therefore, this endeavour does not depend upon any translation of the water in respect to ambient bodies, nor can true circular motion be defined by such translation....; but relative motions...are altogether destitute of any real effect.... It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from the apparent; because the parts of that immovable space in which these motions are performed, do by no means come under the observations of our senses.

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The argument that the motion is absolute, not relative, is incomplete, as it limits the participants relevant to the experiment to only the pail and the water, a limitation that has not been established. In fact, the concavity of the water clearly involves gravitational attraction, and by implication the Earth also is a participant. Here is a critique due to Mach arguing that only relative motion is established:

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Newton's experiment with the rotating vessel of water simply informs us that the relative rotation of the water with respect to the sides of the vessel produces no noticeable centrifugal forces, but that such forces are produced by its relative rotations with respect to the mass of the earth and other celestial bodies.

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The degree in which Mach's hypothesis is integrated in general relativity is discussed in the article Mach's principle ; it is generally held that general relativity is not entirely Machian.

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All observers agree that the surface of rotating water is curved. However, the explanation of this curvature involves centrifugal force for all observers with the exception of a truly stationary observer, who finds the curvature is consistent with the rate of rotation of the water as they observe it, with no need for an additional centrifugal force. Thus, a stationary frame can be identified, and it is not necessary to ask "Stationary with respect to what?":

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The original question, "relative to what frame of reference do the laws of motion hold?" is revealed to be wrongly posed. For the laws of motion essentially determine a class of reference frames, and (in principle) a procedure for constructing them.

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A supplementary thought experiment with the same objective of determining the occurrence of absolute rotation also was proposed by Newton: the example of observing two identical spheres in rotation about their center of gravity and tied together by a string. Occurrence of tension in the string is indicative of absolute rotation; see Rotating spheres.

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The historic interest of the rotating bucket experiment is its usefulness in suggesting one can detect absolute rotation by observation of the shape of the surface of the water. However, one might question just how rotation brings about this change. Below are two approaches to understanding the concavity of the surface of rotating water in a bucket.

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The shape of the surface of a rotating liquid in a bucket can be determined using Newton's laws for the various forces on an element of the surface. For example, see Knudsen and Hjorth. The analysis begins with the free body diagram in the co-rotating frame where the water appears stationary. The height of the water h = h (r) is a function of the radial distance r from the axis of rotation Ω, and the aim is to determine this function. An element of water volume on the surface is shown to be subject to three forces: the vertical force due to gravity F g, the horizontal, radially outward centrifugal force F Cfgl, and the force normal to the surface of the water F n due to the rest of the water surrounding the selected element of surface. The force due to surrounding water is known to be normal to the surface of the water because a liquid in equilibrium cannot support shear stresses. To quote Anthony and Brackett:

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The surface of a fluid of uniform density..., if at rest, is everywhere perpendicular to the lines of force; for if this were not so, the force at a point on the surface could be resolved into two components, one perpendicular and the other tangent to the surface. But from the nature of a fluid, the tangential force would set up a motion of the fluid, which is contrary to the statement that the fluid is at rest.

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Moreover, because the element of water does not move, the sum of all three forces must be zero. To sum to zero, the force of the water must point oppositely to the sum of the centrifugal and gravity forces, which means the surface of the water must adjust so its normal points in this direction. (A very similar problem is the design of a banked turn, where the slope of the turn is set so a car will not slide off the road. The analogy in the case of rotating bucket is that the element of water surface will "slide" up or down the surface unless the normal to the surface aligns with the vector resultant formed by the vector addition F g + F Cfgl.)

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As r increases, the centrifugal force increases according to the relation (the equations are written per unit mass):

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where Ω is the constant rate of rotation of the water. The gravitational force is unchanged at

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where g is the acceleration due to gravity. These two forces add to make a resultant at an angle φ from the vertical given by

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which clearly becomes larger as r increases. To ensure that this resultant is normal to the surface of the water, and therefore can be effectively nulled by the force of the water beneath, the normal to the surface must have the same angle, that is,

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leading to the ordinary differential equation for the shape of the surface:

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