Binding energy in atoms and
nuclei
[Sec. 4.1 Dunlap]
CONCEPT OF BINDING ENERGY
The binding energy of an atom is the energy released as all the constituent
particles (n, p and e) come together FROM INFINITY under both the STRONG
force and the EM force.
The binding energy is something that is LOST from the atomic system. Thus it
is not something that the system possesses.
CALCULATION OF BINDING ENERGY
Total Energy
Total Energy
 Nm

X c
B.E  Zm  Nm  Zm  c  M  X  c
 Zm  Nm  M  X  c
Zm
p
n
 Zm  c
M
2
e
2
p
H
n
n
2
N
A
Z
e
A
Z
A
Z
N
2
 mass constituents - mass atom c 2
2
 B.E
ANOTHER WAY OF VIEWING BINDING ENERGY
+
ATOM
Constituents at infinity
The opposite way of seeing binding energy - is that if B.E.
(MeV) is put into the atom then there is just enough energy
available to split all the constituents of the atoms apart and get
them to rest at infinity.
SINGLE NEUTRON SEPARATION ENERGY
The same method can be used to easily compute the “Single Neutron
Separation Energy” – which is the energy required to “pull” a neutron out of the
nucleus.


Sn  M X N c
 
Sn  M
A
Z
A1
Z
2
 M


A1
Z


X N 1 c  mn c

X N 1  mn  M ZA X N c 2
Note we don’t have to measure Sn directly.
2
2
SINGLE PROTON SEPARATION ENERGY
The same clever strategy applies to finding the “Single Proton Separation
Energy” Sp. But note here there is a difference – we must be careful in
counting electron mass.

Sp  M
A
Z
 
 M 
Sp  M


X N c2  M
A 1
Z 1

 m
A1
Z 1 N
Y
c
YN  m p  me  M
A 1
Z 1
YN
H
M

A
Z
2

XN
A
Z
 mpc2  mec2
XN
 c
 c
2
2
S p  [Mass of Final Products – Mass of Initial atom] c2
ALPHA PARTICLE DECAY ENERGY
In a nuclear decay energy is given out in the separation of particles.
This energy is often referred to as the “Q” of the reaction.
Clearly the Q is the negative of the particle separation energy.

M
Q


 M X   M
  B X   B
A
Z
X N c2  M
A
Z
A
Z
N
N
A4
Z 2 N 2
Y
c  m c  Q
  m c
 B He
A 4
Z 2 N 2
Y
A 4
Z 2 N 2
Y
2
2

2

4

Eq 8.2
Eq. 8.3
Eq. 8.4
235
92
U
CALCULATION OF BINDING ENERGY
Total Energy
Total Energy
 Nm
 M
B.E  Zm  Nm  Zm  c  M 
 Zm  Nm  M  X  c
Zm
p
n
 Zm  c
2
e
2
p
H
n
n
e
A
Z
A
Z
XN
A
Z
XN
c
c
2
 mass constituents - mass atom c 2
2
2
 B.E
Mass Defect
• Mass defect (M.D) is another way of saying nuclear
B.E. It is simply the nuclear B.E. expressed not as
MeV but in mass units (MeV/c2)


B.E  Zm p  Nmn  Zme  c 2  M ZA X N c 2

 
 ZmH  Nmn  M ZA X c 2
 mass constituents - mass atom c 2

M .D.  Zm p  Nmn  Zme   M ZA X N

 

 ZmH  Nmn  M ZA X
= Mass constituents of atom – mass of atom
Mass Excess
• Do not confuse Mass Excess  with Mass Defect
(or Binding Energy). Mass Excess  is just a
CONVENIENT WAY to write down the mass of a
nucleus in amu (u). 1u = 931.5MeV


M X N  Au  
A
Z
This is just a common sense thing. The mass of a nucleus can get
very large if expressed in MeV and will always be approximately
equal to Au because it is made up of A nucleons. It is thus
convenient to tabulate  rather than the whole nuclear mass.
  

  M X  Au c
A
Z
Can either be expressed in u or MeV
2
MeV
Mass Excess – Example on 238U
(238U)= .0507826 u
M(238U)=238+ .0507826 u =238.0507826 u
= 238.0507826 x 931.494 MeV/c2
= 221,742 . 875 MeV/c2
Armed with this information we can work out the B.E. of 238U
Mass Deficit + Binding Energy of
238
92
86,319 . 736 MeV /c2
92 proton mass =
146 neutron mass = 137,174 . 446 MeV /c2
92 electron mass=
47 . 012 MeV /c2
Mass constituents = 223,541 . 194 MeV /c2
M(238U) observed
= 221,742 . 875 MeV/c2
Mass Defect
=
1,798 . 319 MeV/c2
Binding Energy
=
1,798 . 319 MeV
Electronic B.E
=
. 795 MeV
Nuclear B.E.
=
1,797 . 52 MeV
B.E/nucleon
=
1,797.52/238= 7.55MeV
U146
How much is electronic binding energy?
There are two types of binding energy in the atom – Strong Nuclear B.E.
and the Electromagnetic B.E. of the electrons to the nucleus.
B  X   BNuclear  X   BEM  X 
A
Z
A
Z
A
Z
 BNuclear  X   2.08 10
A
Z
5
BEM  238
U

2.08

10
  92 

92
 0.795MeV
7/3
5
Z
7/3
THE FAMOUS B/A (binding energy per nucleon) CURVE