Meta tags:
Headings (most frequently used words):
decay, beta, capture, of, and, transitions, value, electron, nuclides, conservation, neutrinos, types, number, is, conserved, fermi, category, with, qβc, positive, qβb, contents, description, history, nuclear, transmutation, rules, for, energy, release, emission, spectrum, helicity, polarization, electrons, positrons, emitted, in, bound, state, double, see, also, notes, references, further, reading, external, links, discovery, initial, characterization, non, parity, competition, stability, naturally, occurring, baryon, lepton, angular, momentum, function, kurie, plot, gamow, teller, forbidden, first, negative, but, second, low, much, higher,
Text of the page (most frequently used words):
the (477), decay (209), beta (168), and (167), displaystyle (150), #electron (97), for (74), #energy (68), nuclear (57), with (51), that (51), capture (45), text (43), neutron (41), edit (41), nucleus (40), this (38), proton (38), are (38), from (35), mass (35), mathit (34), neutrino (33), doi (32), state (32), into (32), right (30), emission (29), left (28), physics (27), bibcode (27), value (27), number (27), positron (26), nuclides (25), can (25), bound (23), which (23), fermi (22), process (21), atomic (21), momentum (21), nuclei (20), emitted (20), transitions (20), physical (19), particle (19), one (19), have (19), was (18), all (18), stable (18), antineutrino (18), particles (17), half (17), kev (17), known (17), but (17), 187 (16), spin (15), ionized (15), decays (15), quark (15), electrons (15), where (15), not (15), angular (15), spectrum (15), weak (14), retrieved (14), other (14), isbn (14), 1103 (14), review (14), atoms (14), 978 (14), element (14), also (14), total (14), allowed (14), only (14), example (14), double (13), alpha (13), archived (13), first (13), parity (13), neutrinos (13), conservation (13), has (13), neutral (13), transition (12), fully (12), energies (12), original (12), forbidden (12), life (12), positive (12), nuclide (12), binding (12), atom (12), two (11), than (11), while (11), final (11), boson (11), kinetic (11), states (10), gamma (10), radioactive (10), type (10), kurie (10), function (10), its (10), they (10), higher (10), however (10), 163 (10), when (10), between (10), toggle (9), table (9), about (9), see (9), nobel (9), case (9), alvarez (9), quarks (9), since (9), possible (9), change (9), main (9), delta (9), these (9), were (9), initial (9), types (9), down (9), may (8), s2cid (8), arxiv (8), minus (8), physrev (8), 124 (8), energetically (8), undergo (8), given (8), subsection (8), their (8), observed (8), article (8), years (8), negative (8), conserved (8), mev (8), because (8), equation (8), wikipedia (7), non (7), 2026 (7), processes (7), 104 (7), 205 (7), theory (7), plot (7), order (7), prize (7), free (7), rich (7), some (7), changes (7), thus (7), 210 (7), daughter (7), second (7), more (7), leading (7), must (7), helicity (7), frac (7), called (7), odd (7), search (6), reaction (6), 2018 (6), press (6), 1934 (6), journal (6), 2013 (6), 2015 (6), structure (6), rays (6), much (6), will (6), 194 (6), category (6), following (6), very (6), via (6), therefore (6), move (6), history (6), lepton (6), stability (6), same (6), transmutation (6), virtual (6), force (6), curie (6), equal (6), page (5), links (5), articles (5), 2025 (5), interaction (5), radioactivity (5), burning (5), 2012 (5), springer (5), pmid (5), highly (5), wang (5), 1088 (5), pdf (5), nist (5), radium (5), long (5), study (5), polarization (5), elements (5), science (5), der (5), issn (5), tritium (5), chemistry (5), low (5), law (5), rutherford (5), level (5), due (5), 227 (5), 222 (5), occur (5), charge (5), 241 (5), unstable (5), being (5), values (5), difference (5), shell (5), resulting (5), rules (5), few (5), constant (5), sum (5), teller (5), quantum (5), found (5), eta (5), generic (5), isotope (5), generally (5), even (5), there (5), showed (5), contents (4), statements (4), february (4), description (4), international (4), conversion (4), radiation (4), jensen (4), 1911 (4), further (4), neutrinoless (4), spectra (4), ionization (4), form (4), lives (4), references (4), 243 (4), existence (4), 1957 (4), 1007 (4), experiment (4), model (4), theoretical (4), experimental (4), 1956 (4), orbital (4), meitner (4), chadwick (4), displacement (4), matter (4), measurements (4), would (4), zero (4), excited (4), 244 (4), 148 (4), most (4), both (4), usually (4), does (4), 250 (4), theoretically (4), what (4), undergoes (4), chemical (4), created (4), related (4), new (4), simply (4), high (4), accompanied (4), anti (4), couple (4), decaying (4), pauli (4), tau (4), gamow (4), protons (4), isospin (4), after (4), discovery (4), positrons (4), used (4), line (4), maximum (4), captured (4), account (4), greater (4), plus (4), could (4), light (4), release (4), cases (4), either (4), another (4), diagrams (4), feynman (4), evidence (4), continuous (4), distribution (4), hide (4), sidebar (4), view (3), foundation (3), use (3), wayback (3), unsourced (3), german (3), short (3), national (3), stellar (3), spontaneous (3), fission (3), internal (3), university (3), illustrated (3), controversy (3), consensus (3), october (3), 1987 (3), bosch (3), physrevlett (3), letters (3), 2021 (3), chinese (3), evaluation (3), properties (3), bare (3), 134 (3), 2024 (3), 133 (3), jean (3), 2017 (3), ray (3), 387 (3), 106 (3), upper (3), simple (3), mott (3), 1933 (3), strahlen (3), problem (3), radionuclide (3), nucleon (3), effective (3), sigma (3), interactions (3), march (3), chien (3), shiung (3), 105 (3), lee (3), yang (3), luis (3), cowan (3), reines (3), 103 (3), lise (3), fajans (3), displaced (3), emits (3), omitted (3), 149 (3), too (3), determined (3), large (3), matrix (3), any (3), predicted (3), results (3), then (3), impossible (3), measured (3), like (3), single (3), 249 (3), 141 (3), 247 (3), 114 (3), 228 (3), 112 (3), 125 (3), 212 (3), parent (3), examples (3), lists (3), means (3), happens (3), days (3), less (3), 246 (3), 215 (3), 213 (3), 202 (3), 193 (3), 137 (3), above (3), taken (3), particular (3), cdots (3), respectively (3), small (3), referred (3), selection (3), part (3), hat (3), neutrons (3), occurs (3), citation (3), needed (3), associated (3), considered (3), true (3), released (3), overline (3), speed (3), recoiling (3), numbers (3), bar (3), baryon (3), naturally (3), occurring (3), three (3), competition (3), undergoing (3), isobar (3), nucleons (3), create (3), converts (3), diagram (3), awarded (3), aluminium (3), discovered (3), suggested (3), had (3), ellis (3), products (3), bohr (3), soddy (3), becquerel (3), thomson (3), ratio (3), converted (3), composed (3), tools (3), languages (2), contact (2), privacy (2), policy (2), under (2), additional (2), terms (2), using (2), wikimedia (2), inc (2), commons (2), august (2), categories (2), pages (2), different (2), wikidata (2), org (2), photofission (2), photodisintegration (2), chain (2), fusion (2), nucleosynthesis (2), isomeric (2), cluster (2), machine (2), external (2), chicago (2), carsten (2), 9783034884440 (2), reading (2), physrevc (2), 1522 (2), observation (2), 5190 (2), kondev (2), huang (2), naimi (2), audi (2), 1674 (2), 1137 (2), database (2), investigation (2), 024304 (2), 2014 (2), rare (2), 143 (2), tables (2), graphs (2), ran (2), jun (2), zuber (2), collaboration (2), lorex (2), 232701 (2), american (2), jung (2), 1992 (2), 2164 (2), daudel (2), lecoin (2), 1947 (2), 2022 (2), 2002 (2), publishing (2), spectroscopy (2), 386 (2), 1957phrv (2), 124008 (2), 1207 (2), produced (2), several (2), 368 (2), engineering (2), clarendon (2), zeitschrift (2), für (2), physik (2), wiley (2), 471 (2), conference (2), katrin (2), sterile (2), laboratory (2), center (2), 2009 (2), principles (2), 139 (2), 2005 (2), modern (2), ivanov (2), höllwieser (2), troitskaya (2), wellenzohn (2), berdnikov (2), radiative (2), standard (2), strong (2), analysis (2), 113006 (2), com (2), 031 (2), 1413 (2), 254 (2), 1938 (2), 486 (2), 1938phrv (2), 606 (2), 1937 (2), 1935 (2), 3212 (2), orthmann (2), wilhelm (2), 1930 (2), absolute (2), von (2), 1914 (2), accessed (2), periodic (2), place (2), moyne (2), college (2), third (2), elsevier (2), classification (2), 012015 (2), 578 (2), whose (2), close (2), note (2), should (2), negligible (2), current (2), cannot (2), ame2020 (2), greatly (2), notes (2), absorption (2), effect (2), common (2), antineutrinos (2), antiparticles (2), physicists (2), been (2), extremely (2), least (2), 119 (2), 108 (2), 153 (2), 130 (2), 398 (2), 142 (2), 154 (2), 140 (2), 150 (2), 842 (2), 200 (2), darmstadt (2), shortened (2), increase (2), addition (2), similarly (2), 120 (2), 100 (2), 101 (2), 337 (2), larger (2), though (2), almost (2), likewise (2), escape (2), always (2), rarely (2), carried (2), hydrogen (2), yes (2), superallowed (2), forbiddenness (2), corresponds (2), special (2), isobaric (2), similar (2), spins (2), assuming (2), operator (2), vector (2), coupling (2), mathcal (2), relativistic (2), limit (2), modes (2), according (2), mostly (2), handed (2), screws (2), conversely (2), graph (2), developed (2), within (2), narrow (2), range (2), divided (2), axis (2), find (2), betas (2), expression (2), rho (2), hbar (2), sqrt (2), shown (2), described (2), mechanics (2), applied (2), capturing (2), innermost (2), proceed (2), masses (2), detected (2), 000 (2), mode (2), energetic (2), rest (2), defined (2), ranging (2), typical (2), objects (2), parallel (2), hence (2), leptons (2), begin (2), end (2), ell (2), general (2), just (2), here (2), constituent (2), time (2), lived (2), unpaired (2), forming (2), such (2), excess (2), isobars (2), experimentally (2), drip (2), isotopes (2), comes (2), make (2), during (2), inside (2), converting (2), decreased (2), emitting (2), intermediate (2), charged (2), subsequently (2), increased (2), later (2), proved (2), recognition (2), paper (2), product (2), cosmic (2), termed (2), exist (2), artificial (2), contained (2), obtained (2), frederick (2), explained (2), now (2), well (2), band (2), integral (2), unchanged (2), nitrogen (2), 1913 (2), hahn (2), produces (2), 1900 (2), characterization (2), creates (2), transforming (2), pair (2), neither (2), valley (2), portal (2), shows (2), appearance (2), upload (2), file (2), read (2), english (2), norsk (2), bahasa (2), log (2), donate (2), menu (2), add, topic, mobile, cookie, statement, statistics, developers, code, conduct, legal, safety, contacts, disclaimers, available, apply, site, you, agree, registered, trademark, profit, organization, creative, attribution, sharealike, license, rendered, parsoid, last, edited, utc, hidden, webarchive, template, deprecated, format, chem, tags, july, disputed, accuracy, disputes, cs1, language, sources, https, index, php, title, beta_decay, oldid, 1371600374, yale, lux, israel, japan, united, gnd, authority, control, databases, exchange, triple, cno, cycle, deuterium, simulation, filter, live, chart, iaea, 1997, story, tomonaga, bilenky, 2010, 715, 55217197, 10486, 663891, hdl, 1134, s1063779610050035, 2010ppn, 690b, 1001, 1946, 690, takahashi, boyd, mathews, yokoi, 1528, 9954244, 1987phrvc, 1522t, 1996, cosmochronometry, 5193, 10062738, 1996phrvl, 5190b, 030001, abddae, nubase2020, gov, liu, shuo, gao, chao, chang, 2021phrvc, 104b4304l, belli, bernabei, cappella, caracciolo, cerulli, danevich, marco, incicchitti, poda, polischuk, tretyak, baf, crystal, scintillator, contaminated, 118513731, 1140, epja, i2014, 14134, 2014epja, 134b, 1407, 5844, european, meng, ame, 2020, 030003, abddaf, jing, wen, systematic, nucl, 2606, 08963, bai, blaum, boev, brandau, cvetković, dickel, dillmann, dmytriiev, faestermann, forstner, franczak, geissel, gernhäuser, glorius, griffin, gumberidze, haettner, hillenbrand, kienle, korten, kozhuharov, kuzminchuk, langanke, litvinov, menz, morgenroth, nociforo, nolden, pavićević, petridis, popp, purushothaman, reifarth, sanjari, scheidenberger, spillmann, steck, stöhlker, tanaka, trassinelli, trotsenko, varga, weick, woods, yamaguchi, zhang, zhao, e121, december, society, 39714665, 2501, 06029, ions, project, june, 2167, 10046415, 1992phrvl, 2164j, raymond, maurice, marcel, 2019, 1051, jphysrad, 0194700808023800, 238, phys, sur, possibilité, particulier, radioactivité, phénomène, création, byrne, mixing, ckm, unitarity, abele, mund, overview, konopinski, rose, 1966, siegbhan, vol, north, holland, company, frauenfelder, fromco60, 386f, rodejohann, 119158221, 0954, 3899, 2012jphg, 39l4008r, 1206, 2560, 1948, 1948phrv, 1207k, richardson, paxton, 1936, radiations, artificially, substances, limits, shapes, 381, 1936phrv, 368k, schenter, vogel, 1983, approximation, 396, 5307377, osti, 13182, nse83, a17574, 1983nse, 393s, 393, venkataramaiah, gopala, basavaraju, suryanarayana, sanjeeviah, 1985, relation, 364, 250803189, 0305, 4616, 014, 1985jphg, 359v, 359, massey, 34001940, lccn, 1933tac, book, collisions, versuch, einer, theorie, 177, 125763380, bf01351864, 1934zphy, 161f, 161, nave, hyperphysics, kenneth, krane, november, 80553, introductory, mertens, susanne, 13th, topics, astroparticle, underground, taup, 273, 1875, 3892, 1016, phpro, 043, 2015phpro, 267m, 267, procedia, status, prospects, gilbert, thomas, vaia, context, copper, unusual, ame2012, brookhaven, nudat, data, jevremovic, 201, 85608, business, media, 2011, 2nd, 466, 4200, 6471, crc, bahcall, john, january, 1964, 318, 0004, 637x, 1086, 147755, 1964apj, 318b, astrophysical, interiors, loveland, 232, 11532, gauge, hadronic, qed, linear, 1850199, 119088802, 0217, 751x, 1142, s0217751x18501993, 2018ijmpa, 3350199i, 1805, 09702, precision, 119103283, 2470, 0010, physrevd, 2017phrvd, 95k3006i, 1706, 08687, webb, richard, newscientist, physicist, denied, laymon, franklin, cham, 12607, 12608, 6_3, studies, weinstock, maia, scientificamerican, channeling, ada, lovelace, courageous, hero, ambler, hayward, hoppes, hudson, 1415, 1413w, test, 258, 1956phrv, 254l, question, 497, 486a, gallium, 606a, 135, 1937phrv, 134a, 1968, segré, trower, 226, 81304, discovering, selected, works, www, nobelprize, harrison, kruse, mcguire, detection, confirmation, 17796274, 1126, 1956sci, 103c, brown, 1978, idea, 1063, 2995181, 1978pht, 31i, 23b, today, 155, 0044, 3328, bf01339819, 1930zphy, 143m, über, eine, bestimmung, energie, primären, intensitätsverteilung, magnetischen, spektren, 391, 383, verhandlungen, deutschen, physikalischen, gesellschaft, 2000, aaserud, finn, kragh, helge, rüdinger, erik, stuewer, roger, eds, 2023, 7643, 5313, 0348, 8444, birkhäuser, verlag, kasimir, 1887, 1975, best, refers, way, transformed, moved, dispalcement, annunziata, michael, 384874, handbook, trenn, 1976, isis, pais, abraham, reprint, oxford, 851997, inward, forces, world, basdevant, spiro, 01672, fundamentals, cosmology, cottingham, greenwood, 1986, 521, 31960, cambridge, introduction, bijker, santopinto, valence, sea, 118499855, 1742, 6596, 2015jphcs, 578a2015b, 1412, 5559, series, konya, nagy, 391487, radio, uncertain, inconsistent, sign, error, margin, 231, dominated, nonunique, ground, smaller, involve, fifth, determinations, decisively, determine, whether, accelerated, next, pandemonium, powered, fluorescent, lighting, illumination, betavoltaics, emitters, ordinary, own, believe, never, majorana, units, difficult, rarer, effectively, observe, seen, studied, regard, 117, 191, 253, 123, 118, 110, 129, 165, 171, 127, 147, 122, 063, 126, 169, 614, include, among, mathrm, satisfy, ionizing, accelerates, typically, paths, become, exactly, disallowed, natural, decreases, comparison, variation, rates, environment, 145, 740, 115, 858, 109, 887, 898, 660, 066, discussion, details, please, depending, occupied, phenomenon, shells, back, 4570, 291, heavy, ion, research, manner, fail, lying, orbitals, opened, appreciable, minority, four, per, million, body, fails, gain, necessary, remains, essence, off, require, analogue, proceeds, quickly, axial, produce, flip, matrices, running, over, raising, lowering, operators, classified, including, variety, various, carry, otherwise, velocity, reflects, chirality, naively, speaking, driven, material, longitudinal, studying, square, root, lies, certain, plotted, against, straight, accord, intercept, imparted, franz, approximations, literature, approximated, approx, appears, formula, accounts, coulomb, attraction, repulsion, approximating, wavefunctions, spherically, symmetric, analytically, calculated, radius, fine, originally, rae, leads, follows, shape, depends, below, approximately, golden, rule, perturbation, max, reverse, simplifies, analogous, calculation, take, significant, exceeds, twice, giving, cancel, 2m_, equations, words, rearranged, substituting, neglecting, nearly, away, zm_, consider, 1162, 467, 555, 5125, 6147, 5390, 156, 475, 960, 1982, 782, gives, 511, speeds, ultrarelativistic, compared, neglected, around, tens, consideration, fermions, net, assigned, antileptons, leptonic, rightarrow, equiv, projections, textstyle, individual, doesn, flavor, labelled, antiquarks, earth, periods, significantly, 248, age, universe, hours, equally, likely, clearly, former, latter, requirements, radionuclides, favorable, set, introduced, turn, each, said, presents, local, minimum, neighbour, vice, versa, 350, transmute, orange, blue, black, squares, regions, representing, unbroken, diagonal, competing, simultaneous, converse, sufficient, emit, labels, increasing, alphabet, sole, bromine, krypton, conditions, outside, stars, captures, interacts, comprise, indistinguishable, isolated, requires, happen, lower, mother, goes, opposite, depicted, caused, negatively, positively, promoted, fundamental, noticed, postulated, symmetry, preserved, sketched, design, testing, year, coworkers, asymmetrical, surprising, result, overturned, held, assumptions, work, who, female, chen, ning, tsung, dao, discussed, others, went, hideki, yukawa, gian, carlo, wick, bombarded, identical, those, 1932, nature, carl, david, anderson, irène, joliot, frédéric, written, attempted, resolve, conundrum, suggesting, accounting, missing, yet, 1931, renamed, little, italian, published, his, landmark, supposing, annihilated, quanta, rather, detecting, severe, challenge, indirect, observing, recoil, absorbing, finally, directly, minor, modifications, clyde, enrico, wolfgang, famous, letter, 1920, 1927, along, colleagues, established, statistical, sense, might, violated, ruled, out, notion, how, variability, became, acute, principle, niels, nevill, charles, drummond, molecular, leaves, integer, reduced, planck, discuss, dubious, provided, carries, multiple, lines, diffuse, background, offered, hint, magnetic, accurate, appeared, contradiction, validated, calorimetric, 1929, assumed, broad, 160, counters, hans, geiger, spectrometer, james, danysz, otto, 1901, involves, independently, proposed, places, kazimierz, method, cathode, identify, fact, 1896, newly, 1899, separated, emissions, based, penetration, ability, cause, stopped, thin, sheets, whereas, penetrate, millimetres, identified, still, penetrating, paul, villard, ernest, polonium, thorium, pierre, marie, uranium, henri, figure, remaining, far, curve, leaving, showing, division, having, sodium, magnesium, helium, becomes, 700, carbon, conserves, antielectron, muon, depicting, involving, sometimes, included, basic, mediated, inner, consequence, characterized, relatively, times, allows, creation, flavour, fast, transforms, nor, prior, obtain, probability, forms, existing, wigner, walton, szilárd, świątecki, strassmann, rabi, purcell, proca, oppenheimer, oliphant, mayer, lawrence, davisson, skłodowska, cockcroft, blackett, bethe, scientists, lhc, rhic, gluon, plasma, cosmogenic, primordial, supernova, big, bang, astrophysics, spallation, breaking, radiogenic, island, borromean, halo, magic, mirror, isomers, isodiaphers, isotones, initio, interacting, liquid, drop, models, accompanying, inset, depictions, encyclopedia, item, wikiversity, wikiquote, projects, printable, version, download, print, export, switch, legacy, parser, get, url, cite, information, permanent, link, actions, talk, tiếng, việt, oʻzbekcha, ўзбекча, اردو, українська, татарча, tatarça, türkçe, ไทย, тоҷикӣ, தமிழ், svenska, српски, srpski, slovenščina, slovenčina, සිංහල, srpskohrvatski, српскохрватски, русский, română, português, polski, ਪੰਜਾਬੀ, bokmål, nynorsk, nederlands, plattdüütsch, melayu, македонски, latviešu, lietuvių, кыргызча, 한국어, қазақша, ქართული, 日本語, italiano, íslenska, ido, indonesia, հայերեն, magyar, hrvatski, हिन्दी, עברית, galego, gaeilge, français, suomi, فارسی, euskara, eesti, español, esperanto, ελληνικά, deutsch, dansk, català, বাংলা, български, беларуская, تۆرکجه, asturianu, العربية, aragonés, afrikaans, top, personal, recent, community, learn, help, contribute, random, events, navigation, jump, content,
Text of the page (random words):
rgy by the electron and antineutrino because the reaction will proceed only when the q value is positive β decay can occur when the mass of atom a z x is greater than the mass of atom a z 1 x 38 β decay edit the equations for β decay are similar with the generic equation a z x a z 1 x e ν e giving q m n x z a m n x z 1 a m e m ν e c 2 displaystyle q left m_ text n left ce mathit a _ mathit z x right m_ text n left ce mathit a _ mathit z 1 x right m_ text e m_ nu _ text e right 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 displaystyle q left m left ce mathit a _ mathit z x right m left ce mathit a _ mathit z 1 x right 2m_ text e right c 2 because the reaction will proceed only when the q value is positive β decay can occur when the mass of atom a z x exceeds that of a z 1 x by at least twice the mass of the electron 38 electron capture edit 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 a z x e a z 1 x ν e we have q m n x z a m e m n x z 1 a m ν e c 2 displaystyle q left m_ text n left ce mathit a _ mathit z x right m_ text e m_ text n left ce mathit a _ mathit z 1 x right m_ nu _ text e right c 2 which simplifies to q m x z a m x z 1 a c 2 b n displaystyle q left m left ce mathit a _ mathit z x right m left ce mathit a _ mathit z 1 x right right c 2 b_ text n where b n is the binding energy of the captured electron 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 38 beta emission spectrum edit beta spectrum of 210 bi e max q 1 16 mev is the maximum energy 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 39 n t c l t f z t p e q t 2 displaystyle n t c_ l t f z t pe q t 2 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 2 is the total energy p e c 2 m c 2 displaystyle p sqrt e c 2 mc 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 as an example the beta decay spectrum of 210 bi originally called rae is shown to the right fermi function edit 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 40 f z t 2 1 s γ 1 2 s 2 2 p ρ 2 s 2 e π η γ s i η 2 displaystyle f z t frac 2 1 s gamma 1 2s 2 2p rho 2s 2 e pi eta gamma s i eta 2 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 2 z 2 displaystyle s sqrt 1 alpha 2 z 2 η z e 2 e ℏ c p displaystyle eta pm ze 2 e hbar cp for electrons for positrons and ρ r n ℏ displaystyle rho r_ text n hbar for non relativistic betas q m e c 2 this expression can be approximated by 41 f z t 2 π η 1 e 2 π η displaystyle f z t approx frac 2 pi eta 1 e 2 pi eta other approximations can be found in the literature 42 43 kurie plot edit 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 momentum or energy lies within a certain narrow range divided by the fermi function is plotted against beta particle energy 44 45 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 46 helicity polarization of neutrinos electrons and positrons emitted in beta decay edit 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 47 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 48 the higher the velocity energy of the particles the higher their helicity polarization this reflects the absolute chirality selection in w boson decay citation needed types of beta decay transitions edit main article beta decay transition 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 other decay modes which are rare are known as bound state decay and double beta decay fermi transitions edit 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 displaystyle s 0 leading to an angular momentum change δ j 0 displaystyle delta 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 a displaystyle mathcal o _ text f g_ text v sum _ a hat tau _ a pm with g v displaystyle g_ text v the weak vector coupling constant τ displaystyle tau _ pm the isospin raising and lowering operators and a displaystyle a running over all protons and neutrons in the nucleus gamow teller transitions edit 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 displaystyle s 1 leading to an angular momentum change δ j 0 1 displaystyle delta j 0 pm 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 gt g a a σ a τ a displaystyle mathcal o _ text gt g_ text a sum _ a hat sigma _ a hat tau _ a pm with g a displaystyle g_ text a the weak axial vector coupling constant and σ displaystyle sigma the spin pauli matrices which can produce a spin flip in the decaying nucleon forbidden transitions edit main article beta decay transition 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 j l 1 l l 1 δ π 1 l displaystyle delta j l 1 l l 1 delta pi 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 forbiddenness δ j δ π superallowed 0 no allowed 0 1 no first forbidden 0 1 2 yes second forbidden 1 2 3 no third forbidden 2 3 4 yes bound state β decay edit a very small minority of free neutron decays about four per million are two body decays 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 49 in this type of beta decay in essence all of the neutron decay energy is carried off by the antineutrino 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 can in theory occur for neutral atoms as a new bound state is always opened by the decay but rarely is appreciable bound state β decays were predicted by daudel jean and lecoin in 1947 50 and the phenomenon in fully ionized atoms was first observed for 163 dy 66 in 1992 by jung et al of the darmstadt heavy ion research center though neutral 163 dy is stable fully ionized 163 dy 66 undergoes β decay into the k and l shells with a half life of 47 days 51 the resulting nucleus 163 ho 66 is stable only in this almost fully ionized state and will decay via electron capture back into 163 dy in the neutral state likewise while being stable in the neutral state the fully ionized 205 tl 81 undergoes bound state β decay to 205 pb 81 with a half life of 291 33 27 days 52 53 the half lives of neutral 163 ho and 205 pb are respectively 4570 years and 1 70 10 7 years the q value of bound state beta decay of a highly ionized atom q β b displaystyle q_ beta _ b is related to the q value q β c displaystyle q_ beta _ c of beta decay of neutral atoms by q β b k l q β c δ b e b e k l displaystyle q_ beta _ b k l cdots q_ beta _ c delta b_ e b_ e k l cdots 54 where δ b e b n z 1 b n z displaystyle delta b_ e b_ n z 1 b_ n z is the difference of all ionization energies of the element z 1 and the element z and b e k l displaystyle b_ e k l cdots is the binding energy of the electron created via the β decay in the daughter atom depending on the shell k l occupied in particular for a fully ionized atom since b e k displaystyle b_ e k is usually larger than δ b e displaystyle delta b_ e q β b displaystyle q_ beta _ b is usually larger than q β c displaystyle q_ beta _ c first category nuclides with negative q β c value but positive q β b value edit following the discussion above it is possible that a nuclide stable to β decay q β c displaystyle q_ beta _ c being negative can be unstable q β b displaystyle q_ beta _ b being positive when fully or highly ionized theoretically this happens for the following 12 nuclides 148 eu 163 dy 193 ir 194 au 202 tl 205 tl 213 po 215 at 222 rn n 1 244 pu 243 am and 246 bk 57 54 n 2 the following table lists theoretically possible bound state β transitions that are impossible for neutral atoms the q β c displaystyle q_ beta _ c values are taken from ame2020 55 while the electron binding energies are taken from nist 58 see ran wang 2026 for more details 54 please note that the ionization energies for highly ionized atoms are theoretical parent daughter δ b e displaystyle delta b_ e kev b e k displaystyle b_ e k kev transition q β c displaystyle q_ beta _ c kev q β b displaystyle q_ beta _ b kev 148 eu 148 gd 11 6 59 066 5 0 displaystyle 5 to 0 28 0 19 5 163 dy 163 ho 12 5 65 137 5 2 7 2 displaystyle 5 2 to 7 2 2 8 49 8 193 ir 193 pt 15 9 90 660 3 2 1 2 displaystyle 3 2 to 1 2 56 6 18 2 3 2 3 2 displaystyle 3 2 to 3 2 58 2 16 6 3 2 5 2 displaystyle 3 2 to 5 2 70 9 3 9 194 au 194 hg 16 8 95 898 1 0 displaystyle 1 to 0 27 9 51 2 202 tl 202 pb 17 3 101 337 2 0 displaystyle 2 to 0 39 4 44 6 205 tl 205 pb 17 3 101 337 1 2 5 2 displaystyle 1 2 to 5 2 50 6 33 4 1 2 1 2 displaystyle 1 2 to 1 2 52 9 31 1 213 po 213 at 18 4 109 887 9 2 9 2 displaystyle 9 2 to 9 2 74 0 17 5 215 at 215 rn 18 7 112 842 9 2 9 2 displaystyle 9 2 to 9 2 88 0 6 1 222 rn 222 fr 19 2 115 858 0 2 displaystyle 0 to 2 6 0 90 7 244 pu 244 am 22 142 154 0 6 displaystyle 0 to 6 73 1 47 1 243 am 243 cm 24 145 740 5 2 5 2 displaystyle 5 2 to 5 2 6 9 114 8 5 2 7 2 displaystyle 5 2 to 7 2 48 9 72 8 5 2 1 2 displaystyle 5 2 to 1 2 94 3 27 4 5 2 9 2 displaystyle 5 2 to 9 2 100 9 20 8 5 2 3 2 displaystyle 5 2 to 3 2 100 9 20 8 5 2 7 2 displaystyle 5 2 to 7 2 120 9 0 8 246 bk 246 cf 23 153 124 2 0 displaystyle 2 to 0 120 2 9 9 second category nuclides with positive low q β c value and much higher q β b value edit some other nuclides satisfy q β c 0 displaystyle q_ beta _ c 0 so are theoretically β unstable but their q β c displaystyle q_ beta _ c value are very low so that fully ionizing greatly accelerates β decay since the q β b displaystyle q_ beta _ b value is much higher neutral 187 re does undergo β decay with half life 4 12 10 10 years 59 but bosch et al also at darmstadt observed that for fully ionized 187 re 75 this is shortened to only 32 9 years 60 the increase in q value of beta decay typically means that additional decay paths to some excited states of the daughter nucleus become energetically allowed 54 this is exactly what happens in the case of 187 re in addition to the large increase in decay energy 187 re 75 is energetically allowed to undergo β decay to the first excited state in 187 os 75 a process energetically disallowed for natural 187 re n 3 similarly neutral 241 pu undergoes β decay with a half life of 14 3 years but in its fully ionized state the beta decay half life of 241 pu 94 decreases to 4 2 days 61 for comparison the variation of decay rates of other nuclear processes due to chemical environment is less than 1 theoretical examples of nuclides in this category with q β c 200 k e v displaystyle q_ beta _ c 200 mathrm kev include 187 re 194 os 210 pb 212 at 227 ac 228 ra 241 pu 247 cm 250 cm and 249 bk 54 the following table lists theoretically possible bound state β transitions for these nuclides among which some transitions are impossible for neutral atoms parent daughter δ b e displaystyle delta b_ e kev b e k displaystyle b_ e k kev transition q β c displaystyle q_ beta _ c kev q β b displaystyle q_ beta _ b kev 187 re 187 os 15 2 85 614 5 2 1 2 displaystyle 5 2 to 1 2 2 5 72 9 5 2 3 2 displaystyle 5 2 to 3 2 7 3 63 1 194 os 194 ir 15 6 88 114 0 1 displaystyle 0 to 1 96 6 169 1 0 0 displaystyle 0 to 0 53 5 126 0 0 1 displaystyle 0 to 1 14 3 86 8 0 2 displaystyle 0 to 2 12 3 84 8 0 2 displaystyle 0 to 2 15 6 56 9 0 1 displaystyle 0 to 1 42 1 30 4 0 0 displaystyle 0 to 0 47 0 25 5 0 4 displaystyle 0 to 4 50 5 22 0 0 3 displaystyle 0 to 3 52 3 20 2 0 1 displaystyle 0 to 1 64 4 8 1 0 5 displaystyle 0 to 5 64 9 7 6 210 pb 210 bi 17 6 104 133 0 1 displaystyle 0 to 1 63 5 150 0 0 0 displaystyle 0 to 0 17 0 103 5 212 at 212 rn 18 7 112 842 1 0 displaystyle 1 to 0 30 9 125 0 227 ac 227 th n 4 20 125 250 3 2 1 2 displaystyle 3 2 to 1 2 44 8 150 1 3 2 5 2 displaystyle 3 2 to 5 2 35 5 140 8 3 2 3 2 displaystyle 3 2 to 3 2 20 4 125 7 3 2 3 2 display...
|
