archiwum miesiąca: Czerwiec 2006

Trywialny dowód

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Swego czasu pewien wykładowca, bodajże pan Feynman, napisał książkę – podręcznik do fizyki dla studentów – w której przedstawił pewien temat, wraz z różnymi przykładami i wyliczeniami. Nieco później na jednym ze swoich wykładów rozwiązywał zadanie z materiału, który jak pamiętał opracował w swojej książce. Niestety, po zapisaniu wielu linijek wzorów i kilkakrotnym ścieraniu tablicy, zorientował się, że gdzieś popełnił błąd i wynik nie zgadzał się z teorią. Aby nie marnować czasu na powtórne wyliczenia, postanowił od razu podać prawidłową odpowiedź.

— Czy ktoś z państwa — zwrócił się do studentów — nie ma przypadkiem mojej książki?

— Ja mam — odparł jeden ze słuchaczy.

— Może Pan tam sprawdzić jak rozwiązałem ten problem, bo coś mi nie wychodzi?

Student zaczął szukać, a gdy znalazł zwrócił się do Profesora:

— Znalazłem

— I co tam jest napisane na ten temat

— Tutaj jest napisane, że… dowód jest trywialny, Panie Profesorze

Notes on γ decay

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  1. The emission of an energetic quantum of electromagnetic energy, called a γ ray.
  2. Gamma decay often follows beta or alpha decay or occurs after a nuclear reaction leaves a nucleus in an excited state.
  3. Gamma decay takes an excited nucleus, energy E1, to a lower energy state, energy E2
  4. The energy of the γ ray is given by Eγ = E1 – E2, except for a very small energy given as recoil energy of the daughter nucleus.
  5. Gamma rays can be a mixture of different electric and magnetic multipole radiation, but often is dominated by one multipolarity. The decay probabilities are generally larger for electric multipole transitions than for magnetic multipole of the same order. Also, the partial decay constants for each multipolarity decrease rapidly with L, the angular momentum carried by the γ ray. So if different multipolarities can compete in the decay of a particular state, the relative branches will decrease generally in the following order: E1, M1, E2, M2, E3, M3, E4, M4, E5, etc. Usually the decay is dominated by the lowest multipolarity possible, but sometimes, particularly if the losest one is magnetic, there will be a significant admixture of the next highest one on the list. For example M1, E2 admixtures are quite common.
  6. Angular momentum and parity selection rules:
    1. The γ ray carries an angular momentum L and parity (-1)L if electric and (-1)L+1 if magnetic.
    2. L can have values between |Ii – If| and Ii + If.
    3. If the initial and final parities are equal, then M1, E2, M3, E4, M5, etc. will conserve parity. If the parities of the initial and final states are different, then E1, M2, E3, M4, E5, etc. are possible.
    4. The lowest L which conserves angular momentum and parity is generally dominant, as mentioned above.
    5. Mixtures between M1 and E2 are common, and mixing between these two are given in terms of a mixing ratio δ:
      • Fraction of radiation that is E2 = δ2/(1+δ2)
      • Fraction of radiation that is M1 = 1/(1+δ2)
  7. Ways to measure multipolarities and mixing ratios
    1. Align nuclei by a strong magnetic field imposed on a very cold (few mK) source and measure angular distribution of γ rays.
    2. Measure angular correlations between γ rays in cascade.
    3. Measure internal conversion coefficients (defined below) of transitions.
  8. Internal Conversion – the ejection of an orbital electron from the atom by an excited nucleus, resulting in the nucleus going to a lower energy level. Hence, it can compete with γ decay for de-exciting the nuclei.
    1. The ejected electron has an energy Te = Eg – Be, where Be is the binding energy of the orbital electron emitted.
    2. Each γ transition will result in a number of electron lines corresponding to conversion from the different atomic orbitals – K, L, M, N, etc.
    3. Since the binding energies are dependent on the Z, the observation of both internal conversion and γ transitions between the same two states yields the elemental assignment of the nucleus.
    4. The internal conversion coefficient is defined as the ratio of the decay constants for internal conversion and γ decay between the same two states. α = λe/λγ
    5. Thus, the total decay constant is λ = λγ + λe = λγ(1 + α).
    6. Of course, each atomic shell has its own λe and α.
    7. The internal conversion coefficient α is strongly dependent on the transition energy Eγ and the multipolarity. Thus, they are quite useful in determination of the multipolarity.
    8. E0 transitions cannot occur by γ decay but can go by internal conversion. Thus, transitions between two spin 0 states must proceed by internal conversion.
  9. Things we learn about nuclei from gamma and internal conversion electron spectroscopy:
    1. Energy of excited states.
    2. Parity of excited states
      • Initial and final states have same parity if E0, M1, E2, M3, etc.
      • Initial and final states have different parity if E1, M2, E3, etc.
    3. Intensities of transitions
      • Use calibrated detectors to determine number of gammas and conversion electrons.
      • Deduce relative intensities or even absolute intensities.
      • Useful in determining beta-decay intensities to each excited state, and thus the log ft for beta decay to each excited state.
    4. Position states in level scheme by observing γ rays in coincidence.
    5. To assign decays to a particular isotope you need to observe some of these.
      • γ ray – x ray coincidences will identify an element by the characteristic x-ray energy.
      • Element can be deduced by measuring Eγ and Ee for a transition, since the electron binding energy is characteristic of the element.
      • Can also use the energy difference between the K and L lines of a transition to obtain the element identification.
      • Need to use a mass separator, or perhaps an excitation function, to determine the mass number A.
    6. Absolute decay probabilities can sometimes be measured and these depend on the nuclear structure of the excited states of the nucleus. Useful in testing various nuclear models.