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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 8 Krane 3.9
Atomic masses from:
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 9 Show that
HW 10 Similarly, find Sp and S.
HW 11 Krane 3.13            HW 12 Krane 3.14
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007                               1
(Saed Dababneh).
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, 1st Semester, 2006-2007   2
(Saed Dababneh).
Nuclear Binding Energy
Surface effect                                Coulomb effect

~200 MeV

Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007                    3
(Saed Dababneh).
Nuclear Binding Energy
HW 13
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, 1st Semester, 2006-2007   4
(Saed Dababneh).
Nuclear Binding Energy

Is the nucleon bounded equally to every
other nucleon?
C ≡ this presumed binding energy.
Btot = ½ CA(A-1)
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, 1st Semester, 2006-2007   5
(Saed Dababneh).
Nuclear Binding Energy

Neutron Separation Energy Sn (MeV)
For constant Z
Sn (even N) > Sn (odd N)
For constant N
Sp (even Z) > Sp (odd Z)
Remember HW 12 (Krane 3.14).

208Pb  (doubly magic) 
can then easily remove
the “extra” neutron in
209Pb.

Neutron Number N
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007                                             6
(Saed Dababneh).
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, 1st Semester, 2006-2007   7
(Saed Dababneh).
Neutron Excess

Remember HWc 1.

Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007   8
(Saed Dababneh).
Abundance Systematics
Odd N         Even N   Total
HWc 1\                   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, 1st Semester, 2006-2007              9
(Saed Dababneh).
Abundance Systematics

Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007   10
(Saed Dababneh).
Abundance Systematics

NEUTRON CAPTURE
CROSS SECTION
Formation process
NEUTRON NUMBER

Abundance
ABUNDANCE

r s         r s

MASS NUMBER
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007                    11
(Saed Dababneh).
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007   12
(Saed Dababneh).
The Semi-empirical Mass Formula
• von Weizsäcker in 1935.
• Liquid drop.
• Main assumptions:
1. Incompressible matter of the nucleus 
R  A⅓.
2. Nuclear force saturates.
• Binding energy is the sum of terms:
1. Volume term.                       4. Asymmetry term.
2. Surface term.                      5. Pairing term.
3. Coulomb term.                      6. Closed shell term.

Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007                     13
(Saed Dababneh).
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, 1st Semester, 2006-2007   14
(Saed Dababneh).
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:
2
4r A  2       3          2
0
 4A       3   Nucleons.
ro2
Bs   1
 1
A   A 3
Remember t/R  A-1/3
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007         15
(Saed Dababneh).
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:                                                 4r dr
2

W  Z(Z-1) / R.
• BC / A =
- aC Z(Z-1) / A4/3                                        4 3
r 
3

Remember HW 7 … ?!
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007             16
(Saed Dababneh).
The Semi-empirical Mass Formula

Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007   17
(Saed Dababneh).
The Semi-empirical Mass Formula
HW 14
Show that from our information so                            far we can write:
2              1
M ( A, Z )  AM n  Z ( M n  M H )  aV A  aS A  aC Z ( Z  1) A   3               3
 ...

For A = 125, what value of Z makes M(A,Z) a minimum?

Is this reasonable…???

So …..!!!!
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007                            18
(Saed Dababneh).
The Semi-empirical Mass Formula
Asymmetry Term Ba = - aa (A-2Z)2 / A
• Light nuclei: N = Z = A/2 (preferable).
• Deviation from this “symmetry”  less BE and stability.
• Neutron excess (N-Z) is necessary for heavier nuclei.
• Fraction affected = |N-Z| / A
• Total decrease in BE  fraction x excess.
• Ba / A = - aa (N-Z)2 / A2.
• Back to this when we talk about
the shell model.

Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007   19
(Saed Dababneh).
The Semi-empirical Mass Formula
Pairing Term Bp = 
Extra Binding between pairs of identical nucleons in the same state (Pauli !)
 Stability (e.g. -particle, N=2, Z=2).
even-even more stable than even-odd or odd-even and these are more tightly
bound than odd-odd nuclei.
Remember HWc 1\ ….?!
Bp expected to decrease with A; effect of unpaired nucleon decrease with
total number of nucleons. But empirical evidence show that:
  A-¾ .
 a A 3 4              evenN         evenZ            Effect on:
 p
                                                       • Fission.
  0                       oddA                          • Magnetic moment.
 a A 3 4                                             Effect of high angular
 p

oddN           oddZ            momentum.

Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007                        20
(Saed Dababneh).
The Semi-empirical Mass Formula
Closed Shell Term Bshell = 
• Extra binding energy for magic numbers
of N and Z.
• Shell model.
• 1 – 2 MeV more binding.

Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007   21
(Saed Dababneh).
The Semi-empirical Mass Formula
• Fitting to experimental data.
• More than one set of constants av, as …..
• In what constants does r0 appear?
• Accuracy to ~ 1% of experimental values (BE).
• Atomic masses 1 part in 104.
• Uncertainties at magic numbers.

Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007   22
(Saed Dababneh).

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