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Nuclear Binding Energy
Btot(A,Z) = [ ZmH + Nmn - m(A,Z) ] c2 Bm
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
XY+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)
Lead isotopes Z = 82
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
4r 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: 4r 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.
• Additional term for deformation.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 22
(Saed Dababneh).
