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Nuclear Binding Energy
Btot(A,Z) = [ ZmH + Nmn - m(A,Z) ] c2
Bm
Bave(A,Z) = Btot(A,Z) / A
HW 9 Krane 3.9
Atomic masses from:
HW 10 Krane 3.12
http://physics.nist.gov/cgi-bin/Compositions/stand_alone.pl?ele=&all=all&ascii=ascii&isotype=all
Separation Energy
Neutron separation energy: (BE of last neutron)
Sn = [ m(A-1,Z) + mn – m(A,Z) ] c2
= Btot(A,Z) - Btot(A-1,Z)  HW 11 Show that
HW 12 Similarly, find Sp and S.
Magic
HW 13 Krane 3.13 HW 14 Krane 3.14 numbers
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
1
Nuclear Binding Energy
Magic
numbers
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
2
Nuclear Binding Energy
In general
XY+a
Sa(X) = (ma + mY –mX) c2
= BX –BY –Ba
The energy needed to remove a nucleon from a
nucleus ~ 8 MeV  average binding energy per nucleon
(Exceptions???).
Mass spectroscopy  B.
Nuclear reactions  S.
Nuclear reactions  Q-value
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
3
Nuclear Binding Energy
Surface effect
Coulomb effect
~200 MeV
HWc 4
Think of a computer program to
reproduce this graph.
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
4
Nuclear Binding Energy
HW 15
A typical research reactor has power on the
order of 10 MW.
a) Estimate the number of 235U fission events
that occur in the reactor per second.
b) Estimate the fuel-burning rate in g/s.
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
5
Nuclear Binding Energy
Is the nucleon bounded equally to every
other nucleon?
C ≡ this presumed binding energy.
Btot = C(A-1)  A  ½
Bave = ½ C(A-1) Linear ??!!! Directly proportional ??!!!
Clearly wrong … !  wrong assumption
 finite range of strong force,
and force saturation.
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
6
For constant Z
Sn (even N) > Sn (odd N)
For constant N
Sp (even Z) > Sp (odd Z)
Remember HW 14 (Krane 3.14).
208Pb
(doubly magic) 
can then easily remove
the “extra” neutron in
209Pb.
Neutron Separation Energy Sn (MeV)
Nuclear Binding Energy
Lead isotopes Z = 82
Neutron Number N
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
7
Nuclear Binding Energy
Extra Binding between pairs of “identical” nucleons in the same
state (Pauli … !)  Stability (e.g. -particle, N=2, Z=2).
Sn (A, Z, even N) – Sn (A-1, Z, N-1)
This is the neutron pairing energy.
even-even more stable than even-odd or odd-even and these
are more tightly bound than odd-odd nuclei.
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
8
Abundance Systematics
Odd N
HWc 1\
Even N
Total
Odd Z
Even Z
Total
Compare:
• even Z to odd Z.
• even N to odd N.
• even A to odd A.
• even-even to even-odd to odd-even to odd-odd.
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
9
Neutron Excess
Z Vs N (For Stable Isotopes)
90
Remember HWc 1.
80
70
Z=N
60
Z
50
40
30
20
Odd A
10
Even A
0
0
20
40
60
N
80
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
100
120
140
10
Neutron Excess
Remember HWc 1.
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
11
Abundance Systematics
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
12
ABUNDANCE
Formation process

Abundance
NEUTRON CAPTURE
CROSS SECTION
Abundance Systematics
NEUTRON NUMBER
r s
r s
MASS NUMBER
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
13
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
14
The Semi-empirical Mass Formula
• von Weizsäcker in 1935.
• Liquid drop. Shell structure.
• Main assumptions:
1. Incompressible matter of the nucleus 
R  A⅓.
2. Nuclear force saturates.
• Binding energy is the sum of terms:
1. Volume term.
2. Surface term.
3. Coulomb term.
…..
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
4. Asymmetry term.
5. Pairing term.
6. Closed shell term.
15
The Semi-empirical Mass Formula
Volume Term Bv = + av A
Bv  volume  R3  A  Bv / A is a constant
i.e. number of neighbors of each nucleon is
independent of the overall size of the nucleus.
BV
 constant
A
The other terms
are “corrections” to
this term.
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
16
The Semi-empirical Mass Formula
Surface Term Bs = - as A⅔
• Binding energy of inner nucleons is higher than that at the surface.
• Light nuclei contain larger
number (per total) at the surface.
• At the surface there are:
4r A
ro2
2
0
2
3
 4A
2
3
Nucleons.
Bs
1
 1
A A3
Remember t/R  A-1/3
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
17
The Semi-empirical Mass Formula
Coulomb Term BC = - aC Z(Z-1) / A⅓
• Charge density   Z / R3.
• W  2 R5. Why ???
• W  Z2 / R.
• Actually:
W  Z(Z-1) / R.
• BC / A =
- aC Z(Z-1) / A4/3
4r 2 dr
4 3
r 
3
Remember HW 8 … ?!
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
18
The Semi-empirical Mass Formula
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
19
The Semi-empirical Mass Formula
Quiz 1
From our information so
far we can write:
M ( A, Z )  AM n  Z ( M n  M H )  aV A  a S A
2
3
 aC Z ( Z  1) A
1
3
 ...
For A = 125, what value of Z makes M(A,Z) a minimum?
Is this reasonable…???
So …..!!!!
Nuclear and Radiation Physics, BAU, First Semester, 2007-2008
(Saed Dababneh).
20

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