WOLFRAM|DEMONSTRATIONS PROJECT

Nuclear Liquid-Drop Model Applied to Radioactive Decay Modes

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atomic number Z
88
mass number A
226
show experimental data
The liquid-drop model in nuclear physics was originally proposed by George Gamow and developed by Hans Bethe and Carl von Weizsäcker in the 1930s. It treats the nucleus as an incompressible fluid of protons and neutrons bound together by the strong nuclear force. It treats the nucleus as an incompressible fluid of protons and neutrons bound together by the strong nuclear force. For a nuclide
A
Z
X
N
containing
Z
protons and
N=A-Z
neutronsWeizsäcker's semi-empirical formula for the mass of a nucleus has the form
M(Z,A)=Z
m
p
+(A-Z)
m
n
-
E
B

2
c
with
E
B
=
a
V
A-
a
S
2/3
A
-
a
C
Z(Z-1)
1/3
A
-
a
A
2
(A-2Z)
A
+
a
P
δ
1/2
A
.
Here
m
p
and
m
n
are the rest mass of the proton and neutron, respectively, in atomic mass units (amu) and
E
B
is the total binding energy, expressed in MeV. The latter is approximated as a sum of five contributions, in the order written: a volume term proportional to
A
, a surface-tension term proportional to the area
2/3
A
, a Coulomb-repulsion term, an asymmetry term, and a pairing term. The coefficients
a
X
are empirically determined, with best current values given here. In the pairing term,
δ=+1
for even-even nuclei, -1 for odd-odd nuclei, and 0 for odd-even nuclei.
In this Demonstration, theoretical values of nuclear masses for all observed nuclides from
Z=4
to 94, calculated to 0.001 amu, are displayed in the boxes. To further extend the liquid drop model, any possible
α
,
-
β
, or
+
β
radioactive decay modes are predicted. Results from the liquid-drop model can be compared with experimental isotope data sources accessible from the Wolfram website. Nuclear masses are generally accurate within .01 amu of experimental values while decay modes are usually about 80% correct. Particularly for heavier nuclei, decay modes including spontaneous fission (SF), cluster emission (of
20
Ne,
28
Mg, etc.), and double beta decay (2
-
β
) are not accounted for by this version of the liquid-drop model.