Тяжёлые адроны ФИАН / 2
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β decay spectrum
Max electron energy
Draw the sqrt(N) spectrum in case mν=0 and !=0
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Parity
Positive parity |
Negative parity |
Q: why no other options?
Spherical coordinates:
Q: why? |
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Parity
Spherical coordinates:
For a state with definite orbital angular momentum |
QM |
WHY?
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Parity
In additional to parity associated with spatial wave functions, the intrinsic wave function of a particle can also have a definite parity, related to the internal structure of the particle.
Sometimes it can be known from theory/structure of the particle, otherwise it has to be measured (assuming parity conservation)
For fermions, e(antiparticle) = − e (particle)
[from the structure of Dirac equation]
For bosons, e(antiparticle) = e (particle)
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Parity
Sometimes, parity cannot be determined
(e.g. if some particles are only produced in pairs)
We have to make some convention
and measure the rest of the particles from the group
Conventions:
Fermion parity is positive
Baryon parity is positive
Strange particle parity is positive
So, for some set of particles the parity was “agreed on”, the parity of others |
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were measured/calculated (e.g. nuclei) |
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π+ parity
1.The pion is captured by a deuteron nucleus. Since the pion is more massive than an electron by a factor of 250, the radius of a bound state is much smaller than that for an electron. Recall that the Bohr radius is 4πε0hbar2/mee2 for an electron; for a π- it is the same except to replace me by the reduced mass of the pion-deuteron system. Hence when a slow pion approaches a nucleus it can easily replace the electron, forming a mesic-deuterium.
2.As the pion drops to lower energy bound states it emits X-rays. The emitted X-rays tell us that the π- settles into an S-state before interacting.
3.The total angular momentum of the initial state must be J=1 since the pion is spin 0, the deuteron is spin 1, and the orbital angular momentum is 0.
4.The final state consists of two non-relativistic neutrons. Therefore we can write a wave function for their state, with space and spin separated (only possible in the non-relativistic limit):ψ = φ(space) α(spin) = φ(r) α(S, Sz)
5.For a state consisting of two spin ½ neutrons, the possible values of combined spin are S=0 or 1.
6.There are 3 substates for S=1 and 1 for S=0. Using the ket notation |S, Sz> (different from the text) and denoting the two neutrons as |Sz1, Sz2>, the four states are:
The S=1 states are symmetric upon exchange of particles, while the S=0 state |
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is anti-symmetric. Thus the exchange symmetry of the spin function is (-1)S+1. |
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π+ parity
7.The space function can be further decomposed into a radial part and an angular part -- the potential between the neutrons depends on their relative separation only. Therefore, the exchange symmetry of the spatial part of the wave function is (-1)L.
8.The overall exchange symmetry of the final state wavefunction is (-1)L+S+1.
9.Since the state consists of two fermions, the overall exchange symmetry must be negative. Therefore L+S must be even.
10.Since the initial state is J=1, so must be the final state (conservation of angular momentum).
11.Given that S=0 or 1, a state with J=1 can be formed by combining states with {L=0, S=1}, {L=1, S=0}, {L=1, S=1}, or {L=2, S=1}. Of these, only the combination of {L=1, S=1} also satisfies the requirement that L+S be even.
12.Thus the two neutrons must be in a state with L=1 and S=1 which has parity (-1)L = -1
13.The parity of the neutrons is positive, thus the parity of the final state is -1.
14.The parity of the initial state is the parity of the pion times the parity of the deuteron. The deuteron is a neutron and proton state with zero orbital angular momentum and symmetric spin wave function. The parity of the deuteron is positive. Therefore, the parity of the pion must be negative.
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π0 parity
π0→ γγ decay, with 2 converted photons, measuring the planes of conversions (correlated to photon spin)
P(π0) = −1
Parity of the Neutral Pion and the Decay pi0 --> 2e++2e- -
INSPIRE (inspirehep.net)
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Fermi and Gamov-Teller rules (β decays)
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Relativistic case
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