PHY431 Lecture 3
1
2/11/2000
5.1.
The Measured Nuclear Binding Energy
The binding energy per
nucleon, derived from the
measurement
of
the
atomic masses, is shown
as function of the atomic
number
A
in the figure.
Clearly, for atomic num-
ber below about 30, a
smooth curve is not ob-
served, and the peaks and
dips indicate more struc-
ture than the naive liquid
drop model can explain.
The distribution peaks
around
A
=59 for Co and
Fe, and drops off on either
side of that. Indeed Fe is
the most stable element
around.
Of course, for a given atomic number
A
, a variety of stable and radioactive elements with
N+Z=A
ex-
ist. This is the so-called valley of stability, be-
cause in the neutron number
N
versus proton
number
Z
plot, the stable elements form a nar-
row valley somewhat above the line
N=Z
; see
the figure beside. The further one departs from
the center of the valley, the more unstable the
elements encountered.
From this figure, and for the detailed study of
the binding energies, it appears that at certain
values for
N
or
Z
nuclei are particularly stable.
These numbers have been historically known
as
magic numbers
, and are
N
or
Z
= 2, 8, 20,
28, 50, 82, and 126. The existence of the magic
numbers was finally explained in the frame-
work of the nuclear shell model: at certain fill-
ings the low lying set of shells are fully filled,
and the next lowest empty energy level is sepa-
rated from the filled shell by a relatively large
energy gap.
The details of the energy level structure depend
crucially on the details of the (strong) nuclear
potential. A naive harmonic potential of
square-well potential are unable to explain the
existence of the magic numbers. The existence
Binding energy per
nucleon, for all stable
nuclei up to A=21,
and for a stable nu-
cleus of A=29, 39, 49
etc. above. Data from
Wapstra and Audi
(1985).
The
β
-stability valley. Black squares
are the natural stable nuclei, and
neighbors between the drip lines are
radioactive, often artificial.
Data:
Chart of
Nuclides
, 1977,
GE Company.
proton drip line
neutron drip line
28.3
/
4
MeV+6.25 keV
×
A