Fermi decay

In the beta-decay allowed approximation, we neglect the variation of the lepton wave-functions over the nuclear volume and the nuclear momentum (this is equivalent to neglecting all total lepton orbital angular momenta L > 0). The total angular momentum carried off by the leptons is their total spin i.e. 5 = 1 or 0, since each lepton has When the lepton spins in the final state are antiparallel, se+s = stot = 0 the process is the Fermi transition with Vector coupling constant g = Cv (e.g. a pure Fermi decay 140(J r = 0+) —>14 N(JJ = 0+)). When the final state lepton spins are parallel, se + sv = stot = 1> the process is... 

Pauli proposed that two particles were emitted, and Fermi called the second one a neutrino, V. The complete process therefore is n — p -H e 9. Owing to the low probabiHty of its interacting with other particles, the neutrino was not observed until 1959. Before the j3 -decay takes place there are no free leptons, so the conservation of leptons requires that there be a net of 2ero leptons afterward. Therefore, the associated neutrino is designated an antineutrino, 9-, that is, the emitted electron (lepton) and antineutrino (antilepton) cancel and give a net of 2ero leptons. 

However, it has been pointed out 13 16> for large organic molecules ( statistical limit case) that the decay times and quantum yields can legitimately be handled by the Fermi golden rule ... 

Quantitative data on local structure can be obtained via an analysis of the decaying slope next to the absorption edge. The absorption of an X-ray photon boosts a core electron up into an unoccupied band of the material which, in a metal, is the conduction band above the Fermi level. Electrons in such a band behave as if nearly free and no fine structure would be expected on the absorption tail . However, fine structure is observed up to 500 to 1000eV above the edge (see Figure 2.73(b)). The ripples are known as the Kronig fine structure or extended X-ray absorption fine structure (EX AFS). 

The actual curve, however, is somewhat modified by Coulomb interaction between the electron or positron and the nucleus. This is allowed for by multiplication with a dimensionless function F(Z p), which leads to a correction factor / for the total decay rate, and it is the product ft that is used for purposes of comparing measured lifetimes with theory. The most rapid decays, with ft = 103 to 104 s, are known as super alio wed . These include 0+ to 0+ decays having A//, 2 = 2 and ft is found experimentally to be close to 3000 s, giving the coupling constant for the Fermi interaction... 

Consider two different metals in contact and assume that both are well described by the Thomas-Fermi model (see Problem 3.3) with a decay length of Ltf 0.5 A. (a) Calculate the dipole potential drop at the contact if both metals carry equal and opposite charges of 0.1 C m 2. (b) If the work functions of the two metals differ by 0.5 eV, how large is the surface-charge density on each meted ... 

Fermi realized this meant that, if uranium, the heaviest known element, was irradiated with neutrons, it might decay to form a previously unknown transuranic element. Uranium has an atomic number of 92 beta decay would convert it to element 93 , anew member of the Periodic Table. 

The first genuine transuranic element was discovered at Berkeley, where Edwin McMillan used Lawrence s cyclotron in 1939 to bombard uranium with slow neutrons. He saw beta decay from what he predicted was element 93, and set about trying to isolate it. McMillan saw that the element sits beneath the transition metal rhenium in the Periodic Table, and so he assumed it should share some of rhenium s chemical properties. But when he and Fermi s one-time collaborator Emilio Segre performed a chemical analysis, they found that eka-rhenium (in Mendeleyev s terminology) behaved instead like a lanthanide, the series of fourteen elements that loops out of the table after lanthanum (see page 152). Disappointed, they figured that all they had found was one of these known elements. 

For the Boltzmann formulation we introduce, for each point on the Fermi surface, the relaxation time t, defined so that any disturbance 5/ of the equilibrium Fermi function / decays according to the law... 

Another way of showing that p behaves as T2 is to note that the rate of decay due to the Auger effect of an electron with energy above the Fermi energy is (Ziman 1964a, p. 415) proportional to (nkBT)2 +ej for the small values of x produced by a field, this gives a time of relaxation proportional to T2 (Hodges et al 1971) these authors find a proportionality with T2 ln (kBT/E ). ... 

Millikan s experiment did not prove, of course, that (he charge on the cathode ray. beta ray, photoelectric, or Zeeman particle was e. But if we call all such particles electrons, and assume that they have e/m = 1.76 x Hi" coulombs/kg. and e = 1.60 x 10" coulomb (and hence m =9.1 x 10 " kg), we find that they fit very well into Bohr s theory of the hydrogen atom and successive, more comprehensive atomic theories, into Richardson s equations for thermionic emission, into Fermi s theory of beta decay, and so on. In other words, a whole web of modem theory and experiment defines the electron. The best current value of e = (1.60206 0.00003) x 10 g coulomb. 

FERMI SELECTION RULES. A set of selection rules for bem decay where g(E) dE is the number of slates in the energy range E to E + dE. 

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