[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki2\/fermis-interaction-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki2\/fermis-interaction-wikipedia\/","headline":"Fermi’s interaction – Wikipedia","name":"Fermi’s interaction – Wikipedia","description":"before-content-x4 Mechanism of beta decay proposed in 1933 \u03b2\u2212 decay in an atomic nucleus (the accompanying antineutrino is omitted). The","datePublished":"2016-03-06","dateModified":"2016-03-06","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki2\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki2\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/a\/aa\/Beta-minus_Decay.svg\/360px-Beta-minus_Decay.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/a\/aa\/Beta-minus_Decay.svg\/360px-Beta-minus_Decay.svg.png","height":"245","width":"360"},"url":"https:\/\/wiki.edu.vn\/en\/wiki2\/fermis-interaction-wikipedia\/","wordCount":21510,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4Mechanism of beta decay proposed in 1933  \u03b2\u2212 decay in an atomic nucleus (the accompanying antineutrino is omitted). The inset shows beta decay of a free neutron. In both processes, the intermediate emission of a virtual W\u2212 boson (which then decays to electron and antineutrino) is not shown.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In particle physics, Fermi’s interaction (also the Fermi theory of beta decay or the Fermi four-fermion interaction) is an explanation of the beta decay, proposed by Enrico Fermi in 1933.[1] The theory posits four fermions directly interacting with one another (at one vertex of the associated Feynman diagram). This interaction explains beta decay of a neutron by direct coupling of a neutron with an electron, a neutrino (later determined to be an antineutrino) and a proton.[2]Fermi first introduced this coupling in his description of beta decay in 1933.[3]  The Fermi interaction was the precursor to the theory for the weak interaction where the interaction between the proton\u2013neutron and electron\u2013antineutrino is mediated by a virtual W\u2212 boson, of which the Fermi theory is the low-energy effective field theory.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsHistory of initial rejection and later publication[edit]The “tentativo”[edit]Definitions[edit]Electron state[edit]Neutrino state[edit]Heavy particle state[edit]Hamiltonian[edit]Matrix elements[edit]Transition probability[edit]Forbidden transitions[edit]Influence[edit]Later developments[edit]Fermi constant[edit]References[edit]History of initial rejection and later publication[edit]Fermi first submitted his “tentative” theory of beta decay to the prestigious science journal Nature, which rejected it “because it contained speculations too remote from reality to be of interest to the reader.[4]” Nature later admitted the rejection to be one of the great editorial blunders in its history.[5] Fermi then submitted revised versions of the paper to Italian and German publications, which accepted and published them in those languages in 1933 and 1934.[6][7][8][9] The paper did not appear at the time in a primary publication in English.[5]  An English translation of the seminal paper was published in the American Journal of Physics in 1968.[9]Fermi found the initial rejection of the paper so troubling that he decided to take some time off from theoretical physics, and do only experimental physics. This would lead shortly to his famous work with activation of nuclei with slow neutrons.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4The “tentativo”[edit]Definitions[edit]The theory deals with three types of particles presumed to be in direct interaction: initially a \u201cheavy particle\u201d in the \u201cneutron state\u201d (\u03c1=+1{displaystyle rho =+1}), which then transitions into its \u201cproton state\u201d (\u03c1=\u22121{displaystyle rho =-1}) with the emission of an electron and a neutrino.Electron state[edit]\u03c8=\u2211s\u03c8sas,{displaystyle psi =sum _{s}psi _{s}a_{s},}where \u03c8{displaystyle psi } is the single-electron wavefunction, \u03c8s{displaystyle psi _{s}} are its stationary states.as{displaystyle a_{s}} is the operator which annihilates an electron in state s{displaystyle s} which acts on the Fock space asas\u03a8(N1,N2,\u2026,Ns,\u2026)=(\u22121)N1+N2+\u22ef+Ns\u22121(1\u2212Ns)\u03a8(N1,N2,\u2026,1\u2212Ns,\u2026).{displaystyle a_{s}Psi (N_{1},N_{2},ldots ,N_{s},ldots )=(-1)^{N_{1}+N_{2}+cdots +N_{s}-1}(1-N_{s})Psi (N_{1},N_{2},ldots ,1-N_{s},ldots ).}as\u2217{displaystyle a_{s}^{*}} is the creation operator for electron state s{displaystyle s}:as\u2217\u03a8(N1,N2,\u2026,Ns,\u2026)=(\u22121)N1+N2+\u22ef+Ns\u22121Ns\u03a8(N1,N2,\u2026,1\u2212Ns,\u2026).{displaystyle a_{s}^{*}Psi (N_{1},N_{2},ldots ,N_{s},ldots )=(-1)^{N_{1}+N_{2}+cdots +N_{s}-1}N_{s}Psi (N_{1},N_{2},ldots ,1-N_{s},ldots ).}Neutrino state[edit]Similarly,\u03d5=\u2211\u03c3\u03d5\u03c3b\u03c3,{displaystyle phi =sum _{sigma }phi _{sigma }b_{sigma },}where \u03d5{displaystyle phi } is the single-neutrino wavefunction, and \u03d5\u03c3{displaystyle phi _{sigma }} are its stationary states.b\u03c3{displaystyle b_{sigma }} is the operator which annihilates a neutrino in state \u03c3{displaystyle sigma } which acts on the Fock space asb\u03c3\u03a6(M1,M2,\u2026,M\u03c3,\u2026)=(\u22121)M1+M2+\u22ef+M\u03c3\u22121(1\u2212M\u03c3)\u03a6(M1,M2,\u2026,1\u2212M\u03c3,\u2026).{displaystyle b_{sigma }Phi (M_{1},M_{2},ldots ,M_{sigma },ldots )=(-1)^{M_{1}+M_{2}+cdots +M_{sigma }-1}(1-M_{sigma })Phi (M_{1},M_{2},ldots ,1-M_{sigma },ldots ).}b\u03c3\u2217{displaystyle b_{sigma }^{*}} is the creation operator for neutrino state \u03c3{displaystyle sigma }.Heavy particle state[edit]\u03c1{displaystyle rho } is the operator introduced by Heisenberg (later generalized into isospin) that acts on a heavy particle state, which has eigenvalue +1 when the particle is a neutron, and \u22121 if the particle is a proton. Therefore, heavy particle states will be represented by two-row column vectors, where(10){displaystyle {begin{pmatrix}1\\0end{pmatrix}}}represents a neutron, and(01){displaystyle {begin{pmatrix}0\\1end{pmatrix}}}represents a proton (in the representation where \u03c1{displaystyle rho } is the usual \u03c3z{displaystyle sigma _{z}} spin matrix).The operators that change a heavy particle from a proton into a neutron and vice versa are respectively represented byQ=\u03c3x\u2212i\u03c3y=(0100){displaystyle Q=sigma _{x}-isigma _{y}={begin{pmatrix}0&1\\0&0end{pmatrix}}}andQ\u2217=\u03c3x+i\u03c3y=(0010).{displaystyle Q^{*}=sigma _{x}+isigma _{y}={begin{pmatrix}0&0\\1&0end{pmatrix}}.}un{displaystyle u_{n}} resp. vn{displaystyle v_{n}} is an eigenfunction for a neutron resp. proton in the state n{displaystyle n}.Hamiltonian[edit]The Hamiltonian is composed of three parts: Hh.p.{displaystyle H_{text{h.p.}}}, representing the energy of the free heavy particles, Hl.p.{displaystyle H_{text{l.p.}}}, representing the energy of the free light particles, and a part giving the interaction Hint.{displaystyle H_{text{int.}}}.Hh.p.=12(1+\u03c1)N+12(1\u2212\u03c1)P,{displaystyle H_{text{h.p.}}={frac {1}{2}}(1+rho )N+{frac {1}{2}}(1-rho )P,}where N{displaystyle N} and P{displaystyle P} are the energy operators of the neutron and proton respectively, so that if \u03c1=1{displaystyle rho =1}, Hh.p.=N{displaystyle H_{text{h.p.}}=N}, and if \u03c1=\u22121{displaystyle rho =-1}, Hh.p.=P{displaystyle H_{text{h.p.}}=P}.Hl.p.=\u2211sHsNs+\u2211\u03c3K\u03c3M\u03c3,{displaystyle H_{text{l.p.}}=sum _{s}H_{s}N_{s}+sum _{sigma }K_{sigma }M_{sigma },}where Hs{displaystyle H_{s}} is the energy of the electron in the sth{displaystyle s^{text{th}}} state in the nucleus’s Coulomb field, and  Ns{displaystyle N_{s}} is the number of electrons in that state; M\u03c3{displaystyle M_{sigma }} is the number of neutrinos in the \u03c3th{displaystyle sigma ^{text{th}}} state, and K\u03c3{displaystyle K_{sigma }} energy of each such neutrino (assumed to be in a free, plane wave state).The interaction part must contain a term representing the transformation of a proton into a neutron along with the emission of an electron and a neutrino (now known to be an antineutrino), as well as a term for the inverse process; the Coulomb force between the electron and proton is ignored as irrelevant to the \u03b2{displaystyle beta }-decay process.Fermi proposes two possible values for Hint.{displaystyle H_{text{int.}}}: first, a non-relativistic version which ignores spin:Hint.=g[Q\u03c8(x)\u03d5(x)+Q\u2217\u03c8\u2217(x)\u03d5\u2217(x)],{displaystyle H_{text{int.}}=gleft[Qpsi (x)phi (x)+Q^{*}psi ^{*}(x)phi ^{*}(x)right],}and subsequently a version assuming that the light particles are four-component Dirac spinors, but that speed of the heavy particles is small relative to c{displaystyle c} and that the interaction terms analogous to the electromagnetic vector potential can be ignored:Hint.=g[Q\u03c8~\u2217\u03b4\u03c8+Q\u2217\u03c8~\u03b4\u03c8\u2217],{displaystyle H_{text{int.}}=gleft[Q{tilde {psi }}^{*}delta psi +Q^{*}{tilde {psi }}delta psi ^{*}right],}where \u03c8{displaystyle psi } and \u03d5{displaystyle phi } are now four-component Dirac spinors, \u03c8~{displaystyle {tilde {psi }}} represents the Hermitian conjugate of \u03c8{displaystyle psi }, and \u03b4{displaystyle delta } is a matrix(0\u22121001000000100\u221210).{displaystyle {begin{pmatrix}0&-1&0&0\\1&0&0&0\\0&0&0&1\\0&0&-1&0end{pmatrix}}.}Matrix elements[edit]The state of the system is taken to be given by the tuple \u03c1,n,N1,N2,\u2026,M1,M2,\u2026,{displaystyle rho ,n,N_{1},N_{2},ldots ,M_{1},M_{2},ldots ,} where \u03c1=\u00b11{displaystyle rho =pm 1} specifies whether the heavy particle is a neutron or proton, n{displaystyle n} is the quantum state of the heavy particle, Ns{displaystyle N_{s}} is the number of electrons in state s{displaystyle s} and M\u03c3{displaystyle M_{sigma }} is the number of neutrinos in state \u03c3{displaystyle sigma }.Using the relativistic version of Hint.{displaystyle H_{text{int.}}}, Fermi gives the matrix element between the state with a neutron in state n{displaystyle n} and no electrons resp. neutrinos present in state s{displaystyle s} resp. \u03c3{displaystyle sigma }, and the state with a proton in state m{displaystyle m} and an electron and a neutrino present in states s{displaystyle s} and \u03c3{displaystyle sigma } asH\u03c1=\u22121,m,Ns=1,M\u03c3=1\u03c1=1,n,Ns=0,M\u03c3=0=\u00b1g\u222bvm\u2217un\u03c8~s\u03b4\u03d5\u03c3\u2217d\u03c4,{displaystyle H_{rho =-1,m,N_{s}=1,M_{sigma }=1}^{rho =1,n,N_{s}=0,M_{sigma }=0}=pm gint v_{m}^{*}u_{n}{tilde {psi }}_{s}delta phi _{sigma }^{*}dtau ,}where the integral is taken over the entire configuration space of the heavy particles (except for \u03c1{displaystyle rho }). The \u00b1{displaystyle pm } is determined by whether the total number of light particles is odd (\u2212) or even (+).Transition probability[edit]To calculate the lifetime of a neutron in a state n{displaystyle n} according to the usual Quantum perturbation theory, the above matrix elements must be summed over all unoccupied electron and neutrino states. This is simplified by assuming that the electron and neutrino eigenfunctions \u03c8s{displaystyle psi _{s}} and \u03d5\u03c3{displaystyle phi _{sigma }} are constant within the nucleus (i.e., their Compton wavelength is much smaller than the size of the nucleus). This leads toH\u03c1=\u22121,m,Ns=1,M\u03c3=1\u03c1=1,n,Ns=0,M\u03c3=0=\u00b1g\u03c8~s\u03b4\u03d5\u03c3\u2217\u222bvm\u2217und\u03c4,{displaystyle H_{rho =-1,m,N_{s}=1,M_{sigma }=1}^{rho =1,n,N_{s}=0,M_{sigma }=0}=pm g{tilde {psi }}_{s}delta phi _{sigma }^{*}int v_{m}^{*}u_{n}dtau ,}where \u03c8s{displaystyle psi _{s}} and \u03d5\u03c3{displaystyle phi _{sigma }} are now evaluated at the position of the nucleus.According to Fermi’s golden rule[further explanation needed], the probability of this transition is|a\u03c1=\u22121,m,Ns=1,M\u03c3=1\u03c1=1,n,Ns=0,M\u03c3=0|2=|H\u03c1=\u22121,m,Ns=1,M\u03c3=1\u03c1=1,n,Ns=0,M\u03c3=0\u00d7exp\u20612\u03c0ih(\u2212W+Hs+K\u03c3)t\u22121\u2212W+Hs+K\u03c3|2=4|H\u03c1=\u22121,m,Ns=1,M\u03c3=1\u03c1=1,n,Ns=0,M\u03c3=0|2\u00d7sin2\u2061(\u03c0th(\u2212W+Hs+K\u03c3))(\u2212W+Hs+K\u03c3)2,{displaystyle {begin{aligned}left|a_{rho =-1,m,N_{s}=1,M_{sigma }=1}^{rho =1,n,N_{s}=0,M_{sigma }=0}right|^{2}&=left|H_{rho =-1,m,N_{s}=1,M_{sigma }=1}^{rho =1,n,N_{s}=0,M_{sigma }=0}times {frac {exp {{frac {2pi i}{h}}(-W+H_{s}+K_{sigma })t}-1}{-W+H_{s}+K_{sigma }}}right|^{2}\\&=4left|H_{rho =-1,m,N_{s}=1,M_{sigma }=1}^{rho =1,n,N_{s}=0,M_{sigma }=0}right|^{2}times {frac {sin ^{2}left({frac {pi t}{h}}(-W+H_{s}+K_{sigma })right)}{(-W+H_{s}+K_{sigma })^{2}}},end{aligned}}}where W{displaystyle W} is the difference in the energy of the proton and neutron states.Averaging over all positive-energy neutrino spin \/ momentum directions (where \u03a9\u22121{displaystyle Omega ^{-1}} is the density of neutrino states, eventually taken to infinity), we obtain\u27e8|H\u03c1=\u22121,m,Ns=1,M\u03c3=1\u03c1=1,n,Ns=0,M\u03c3=0|2\u27e9avg=g24\u03a9|\u222bvm\u2217und\u03c4|2(\u03c8~s\u03c8s\u2212\u03bcc2K\u03c3\u03c8~s\u03b2\u03c8s),{displaystyle leftlangle left|H_{rho =-1,m,N_{s}=1,M_{sigma }=1}^{rho =1,n,N_{s}=0,M_{sigma }=0}right|^{2}rightrangle _{text{avg}}={frac {g^{2}}{4Omega }}left|int v_{m}^{*}u_{n}dtau right|^{2}left({tilde {psi }}_{s}psi _{s}-{frac {mu c^{2}}{K_{sigma }}}{tilde {psi }}_{s}beta psi _{s}right),}where \u03bc{displaystyle mu } is the rest mass of the neutrino and \u03b2{displaystyle beta } is the Dirac matrix.Noting that the transition probability has a sharp maximum for values of p\u03c3{displaystyle p_{sigma }} for which \u2212W+Hs+K\u03c3=0{displaystyle -W+H_{s}+K_{sigma }=0}, this simplifies to[further explanation needed]t8\u03c03g2h4\u00d7|\u222bvm\u2217und\u03c4|2p\u03c32v\u03c3(\u03c8~s\u03c8s\u2212\u03bcc2K\u03c3\u03c8~s\u03b2\u03c8s),{displaystyle t{frac {8pi ^{3}g^{2}}{h^{4}}}times left|int v_{m}^{*}u_{n}dtau right|^{2}{frac {p_{sigma }^{2}}{v_{sigma }}}left({tilde {psi }}_{s}psi _{s}-{frac {mu c^{2}}{K_{sigma }}}{tilde {psi }}_{s}beta psi _{s}right),}where p\u03c3{displaystyle p_{sigma }} and K\u03c3{displaystyle K_{sigma }} is the values for which \u2212W+Hs+K\u03c3=0{displaystyle -W+H_{s}+K_{sigma }=0}.Fermi makes three remarks about this function:Qmn\u2217=\u222bvm\u2217und\u03c4{displaystyle Q_{mn}^{*}=int v_{m}^{*}u_{n}dtau }in the transition probability is normally of magnitude 1, but in special circumstances it vanishes; this leads to (approximate) selection rules for \u03b2{displaystyle beta }-decay.Forbidden transitions[edit]As noted above, when the inner product Qmn\u2217{displaystyle Q_{mn}^{*}} between the heavy particle states un{displaystyle u_{n}} and vm{displaystyle v_{m}} vanishes, the associated transition is “forbidden” (or, rather, much less likely than in cases where it is closer to 1).If the description of the nucleus in terms of the individual quantum states of the protons and neutrons is good, Qmn\u2217{displaystyle Q_{mn}^{*}} vanishes unless the neutron state un{displaystyle u_{n}} and the proton state vm{displaystyle v_{m}} have the same angular momentum; otherwise, the angular momentum of the whole nucleus before and after the decay must be used.Influence[edit]Shortly after Fermi’s paper appeared, Werner Heisenberg noted in a letter to Wolfgang Pauli[10] that the emission and absorption of neutrinos and electrons in the nucleus should, at the second order of perturbation theory, lead to an attraction between protons and neutrons, analogously to how the emission and absorption of photons leads to the electromagnetic force. He found that the force would be of the form Const.r5{displaystyle {frac {text{Const.}}{r^{5}}}}, but that contemporary experimental data led to a value that was too small by a factor of a million.[11]The following year, Hideki Yukawa picked up on this idea,[12] but in his theory the neutrinos and electrons were replaced by a new hypothetical particle with a rest mass approximately 200 times heavier than the electron.[13]Later developments[edit]Fermi’s four-fermion theory describes the weak interaction remarkably well. Unfortunately, the calculated cross-section, or probability of interaction, grows as the square of the energy \u03c3\u2248GF2E2{displaystyle sigma approx G_{rm {F}}^{2}E^{2}}.  Since this cross section grows without bound, the theory is not valid at energies much higher than about 100\u00a0GeV. Here GF is the Fermi constant, which denotes the strength of the interaction. This eventually led to the replacement of the four-fermion contact interaction by a more complete theory (UV completion)\u2014an exchange of a W or Z boson as explained in the electroweak theory.  Fermi’s interaction showing the 4-point fermion vector current, coupled under Fermi’s Coupling Constant GF. Fermi’s Theory was the first theoretical effort in describing nuclear  decay rates for \u03b2 decay.The interaction could also explain muon decay via a coupling of a muon, electron-antineutrino, muon-neutrino and electron, with the same fundamental strength of the interaction. This hypothesis was put forward by Gershtein and Zeldovich and is known as the Vector Current Conservation hypothesis.[14]In the original theory, Fermi assumed that the form of interaction is a contact coupling of two vector currents. Subsequently, it was pointed out by Lee and Yang that nothing prevented the appearance of an axial, parity violating current, and this was confirmed by experiments carried out by Chien-Shiung Wu.[15][16]The inclusion of parity violation in Fermi’s interaction was done by George Gamow and Edward Teller in the so-called Gamow\u2013Teller transitions which described Fermi’s interaction in terms of parity-violating “allowed” decays and parity-conserving “superallowed” decays in terms of anti-parallel and parallel electron and neutrino spin states respectively. Before the advent of the electroweak theory and the Standard Model, George Sudarshan and Robert Marshak, and also independently Richard Feynman and Murray Gell-Mann, were able to determine the correct tensor structure (vector minus axial vector, V \u2212 A) of the four-fermion interaction.[17][18]Fermi constant[edit]The most precise experimental determination of the Fermi constant comes from measurements of the muon lifetime, which is inversely proportional to the square of GF (when neglecting the muon mass against the mass of the W boson).[19] In modern terms, the “reduced Fermi constant”, that is, the constant in natural units is[3][20]GF0=GF(\u210fc)3=28g2MW2c4=1.1663787(6)\u00d710\u22125GeV\u22122\u22484.5437957\u00d71014J\u22122\u00a0.{displaystyle G_{rm {F}}^{0}={frac {G_{rm {F}}}{(hbar c)^{3}}}={frac {sqrt {2}}{8}}{frac {g^{2}}{M_{rm {W}}^{2}c^{4}}}=1.1663787(6)times 10^{-5};{textrm {GeV}}^{-2}approx 4.5437957times 10^{14};{textrm {J}}^{-2} .}Here, g is the coupling constant of the weak interaction, and MW is the mass of the W boson, which mediates the decay in question.In the Standard Model, the Fermi constant is related to the Higgs vacuum expectation valuev=(2GF0)\u22121\/2\u2243246.22GeV{displaystyle v=left({sqrt {2}},G_{rm {F}}^{0}right)^{-1\/2}simeq 246.22;{textrm {GeV}}}.[21]More directly, approximately (tree level for the standard model),GF0\u2243\u03c0\u03b12\u00a0MW2(1\u2212MW2\/MZ2).{displaystyle G_{rm {F}}^{0}simeq {frac {pi alpha }{{sqrt {2}}~M_{rm {W}}^{2}(1-M_{rm {W}}^{2}\/M_{rm {Z}}^{2})}}.}This can be further simplified in terms of the Weinberg angle using the relation between the W and Z bosons with MZ=MWcos\u2061\u03b8W{displaystyle M_{text{Z}}={frac {M_{text{W}}}{cos theta _{text{W}}}}}, so thatGF0\u2243\u03c0\u03b12\u00a0MZ2cos2\u2061\u03b8Wsin2\u2061\u03b8W.{displaystyle G_{rm {F}}^{0}simeq {frac {pi alpha }{{sqrt {2}}~M_{rm {Z}}^{2}cos ^{2}theta _{rm {W}}sin ^{2}theta _{rm {W}}}}.}References[edit]^ Yang, C. N. (2012). “Fermi’s \u03b2-decay Theory”. Asia Pacific Physics Newsletter. 1 (1): 27\u201330. doi:10.1142\/s2251158x12000045.^ Feynman, R.P. (1962). Theory of Fundamental Processes. W. A. Benjamin. Chapters 6 & 7.^ a b Griffiths, D. (2009). Introduction to Elementary Particles (2nd\u00a0ed.). pp.\u00a0314\u2013315. ISBN\u00a0978-3-527-40601-2.^ Pais, Abraham (1986). Inward Bound. Oxford: Oxford University Press. p.\u00a0418. ISBN\u00a00-19-851997-4.^ a b Close, Frank (February 23, 2012). Neutrino. Oxford University Press. Retrieved May 5, 2017.^ Fermi, E. (1933). “Tentativo di una teoria dei raggi \u03b2”. La Ricerca Scientifica (in Italian). 2 (12).^ Fermi, E. (1934). “Tentativo di una teoria dei raggi \u03b2”. Il Nuovo Cimento (in Italian). 11 (1): 1\u201319. Bibcode:1934NCim…11….1F. doi:10.1007\/BF02959820. S2CID\u00a0123342095.^ Fermi, E. (1934). “Versuch einer Theorie der beta-Strahlen. I”. Zeitschrift f\u00fcr Physik (in German). 88 (3\u20134): 161. Bibcode:1934ZPhy…88..161F. doi:10.1007\/BF01351864. S2CID\u00a0125763380.^ a b Wilson, F. L. (1968). “Fermi’s Theory of Beta Decay”. American Journal of Physics. 36 (12): 1150\u20131160. Bibcode:1968AmJPh..36.1150W. doi:10.1119\/1.1974382. Includes complete English translation of Fermi’s 1934 paper in German^ Pauli, Wolfgang (1985). Scientific Correspondence with Bohr, Einstein, Heisenberg a.o. Volume II:1930\u20131939. Springer-Verlag Berlin Heidelberg GmbH. p. 250, letter #341, Heisenberg to Pauli, January 18th 1934.^ Brown, Laurie M (1996). The Origin of the Concept of Nuclear Forces. Institute of Physics Publishing. Section 3.3.^ Yukawa, H. (1935). “On the interaction of elementary particles. I.”. Proceedings of the Physico-Mathematical Society of Japan. 17: 1.^ Mehra, Jagdish (2001). The Historical Development of Quantum Theory, Volume 6 Part 2 (1932\u20131941). Springer. p. 832.^ Gerstein, S. S.; Zeldovich, Ya. B. (1955). “Meson corrections in the theory of beta decay”. Zh. 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