Binding Energy
3.3 Binding Energy
• The binding energy of a nucleus is the
energy required to separate all of the
constituent nucleons from the nucleus so
that they are all unbound and free particles.
• This implies that N
M Z,A
 Zm p  (A  Z)mn 
• And, of course, the mass-energy -N
2
M Z,A c
 Zm p  (A  Z)mn c
2
N
M Z,A
•
is the nuclear mass (no electrons)
• BE defined as --
BE  Zm p  (A  Z)mn c
• BE is always > 0.
2
N
2
 M Z,A c
• To calculate the BE, we do not know
nuclear masses. Therefore, use isotopic
masses --
m p c  m1,1c
2
mn c  mn c
2
2

 mec2


 Eb 

2
Z


N
2
2
2
M Z,A c  M Z ,A c  Z me c   Ebi 


i1
http://www.physics.valpo.edu/physLinks/atomicNuclearLinks.html
• To calculate isotopic masses from  -  MA  M  A
1 uc 2  931.494 MeV ;
 26,56  60.601MeV

MeV 
M 26,56 c   26,56  A 931.494


u 

MeV 
  60.601 MeV  56u 931.494


u 
 52103.065 MeV
2
M 26,56 c 2
c2
52103.065 MeV

 55.934942u


MeV
931.494


u 
Z


2
BE  M Z ,A c  Z me c   E bi 


i1



2
2
2 


 Z m1,1c  Z mec  E b1  A  Z m n c 




2
Z

BE  M Z ,A c   E bi  Z m1,1c 2  Z E b1  A  Z m n c 2
2
i1
Z
BE  M Z ,A c 2  Z m1,1c 2  A  Z m n c 2   E bi  Z E b1
i1
BE  M Z ,A c 2  Z m1,1c 2  A  Z m n c 2

Separation Energies & Systematic Studies
Sn  mZ,A1  mn c  mZ,A c
2
2
S p  mZ1,A  m1,1 c  mZ,Ac
2
2
• Table 3.1 - Can you see any pattern(s)?
• Figure 3.16 - Describe significant features
• Consider the physics that might give rise to
Figure 3.16 -- can we develop a model that would
describe Figure 3.16?
Semi-empirical BE equation
BE

a
v
A
 BE  av A
• Consistent with short-range force; nearly contact
interaction.
• But nucleons on surface are less strongly bound -
BE  R
IF R  A
2
BE  a s A
2
3
• Surface unbinding -
1
3

Semi-empirical BE equation
• Coulomb force from all protons -• This effect can be calculated exactly from electrostatics -
Z(Z 1)
BE  
R
2
3 e Z(Z 1)
BE  
5 4 o Ro A1/ 3
• Coulomb unbinding -

Semi-empirical BE equation
• Systematic studies show that the line of stability moves
from Z = N to N > Z Why?
• Coulomb force demands this -- but -• The asymmetry introduces a nuclear force unbinding --
A - 2Z
2
BE  a sym
A
See next slide
Empirical
Semi-empirical BE equation
Z=N
Z<N
For Z < N, there is
an increased energy
equal to --

N  Z
N  Z 
E increase  
E 

 2
 2 

p
n
p
n
Energy jump
for each proton
# of
protons
N  AZ
A  2Z 2
Eincrease  
 E
 2 
Semi-empirical BE equation
• Systematic studies show like nucleons want to pair and in
pairs are more stable (lower energy) than unpaired.
• Therefore, we add (ad hoc) a pairing energy --
BE  a p A
-3/4
if A & N are even
BE  a p A-3/4 if A & N are odd
BE  0 if A or N is odd


Semi-empirical BE equation
• Combined equation for total BE is -2
(A
2Z)
BE  av A - as A2/3 - a c
- asym
+
1/3
A
A
Z(Z -1)
• Systematic BE data are fit with this function giving -
av , as , asym,  ; ac isknown.
• Using these values of the parameters, one can then
calculate BE for any nuclide (Z,A).

Semi-empirical mass equation
• The isotopic mass of any nucleus can be calculated using
the definition of the BE - but calculating the BE from the
semi-empirical equation:
N
2
M Z,A c 
Zm p  (A  Z)mn c
2
 BE
• And, at constant A, one can find the value of Z at which the
mass is a minimum (Zmin) - (3.30)
• One can also calculate the separation energies.
Semi-empirical mass equation
•
•
•
•
•
•
BE(Z) is a parabolic function of Z at constant A (isobar!)
This curve has a maximum  stability against decay.
The corresponding M Z,A has a minimum at stability.
One curve if A is odd; two curves if A is even. (?)
Separation between the curves is -- 2
With this
semi-empirical model, one can --–
–
–
–
Calculate Q (energy) for decay schemes (, , , , p, n, fission)
Q > 0  decay is possible
Q < 0  decay is not possible
Put semi-empirical mass equation into Excel and calculate all of
the masses in an isobar for a range of Z values.
http://www.physics.valpo.edu/physLinks/atomicNuclearLinks.html