Elements of Modern Physics
360
For
127
I, the odd nucleon is the 53rd proton. Shell model would predict that
it is in the g 7/2 state. However, one finds j = 5/2 for the nucleus. Assuming that
is in the d 5/2 state (see Fig. 9.3), shell model predicts
2 p
e
m
µ ≈
(4.79) whereas
the experimental value is 2 p
e
m
(2.81). The two example, illustrate the usefulness
and limitations of the shell model.
Example 4
A mass spectrometer is used for determining the masses of nuclei. It is based on
the principle that a moving particle subjected to mutually perpendicular electric
and magnetic fields, which are also perpendicular to the velocity of the particle,
is undeviated if
qE + qv × B = 0
(9.127)
or v = | E |/ | B |. If such a particle is now subjected to a magnetic field, it moves
in a circle of radius
r =
p
qB
(9.128)
where p is its momentum. Thus, from the knowledge of v and p, the mass of the
particle can be determined (provided q is known).
Example 5
It is after the case that only a small amount of the target is exposed to a beam of
particles. The reaction produces an unstable isotope which decays. It is of interest
to know the number of unstable nuclei remaining after an exposure to the beam
for time t.
If the target contains N nuclei, the number of reactions per second is
n = NσF
(9.129)
where F is the flux of the beam, i.e. particles/m
2
/s. The net increase dP in the
isotope population over a period dt is
dP = Nσ F dt – λ P dt
(9.130)
where λ is the probability for decay. The solution to this equation is
P(t) = Nσ F(1 – e
– λt
)/λ
(9.131)
For example, consider 1 mg of
23
Na exposed to a neutron beam of
flux 10
14
/cm
2
s. The cross section for the reaction
23
Na(n, γ)
24
Na is about 0.56
barns. Since 1/λ ≈ 21.7 h, and N ≈ 2.6 × 10
19
, the number of isotope nuclei is