16.451 Lecture 21: On to Finite Nuclei! 20/11/2003
Review:
1. Nuclear isotope chart: (lecture 1)
• 304 isotopes with t½ > 109 yrs (age of the earth)
• 177 have even-Z, even-N and J = 0+
• 121 are even-odd and only 6 are odd-odd
• N Z for light nuclei and N > Z for heavy nuclei
Z
N
2. Elastic scattering of electrons: (lecture 7)
• nuclei are approximately spherical
• RMS charge radius fitted to electron scattering data
• mass ~ A, and radius ~ A1/3 so the density M/V constant for nuclei ( 2 x 1017 kg/m3), implying that nuclear matter is like an incompressible fluid
fm3/1A2.1R
1
Recall from lecture 7 -- Nuclear charge distributions from experiment:
fm5.0
fm3/11.1
/)(1
:
a
AR
aRre
formFunctional
o
Approx. constantcentral density
2
continued...
3. Inelastic electron scattering: (lecture 9)
• Excited states can be identified, on a scale of a few MeV above the ground state, e.g.
E (MeV)
3
4. Quantum numbers for nuclear states:
• total angular momentum J, parity (see lecture 12, slide 10 for addition rules)
• isospin, T: (lecture 13)
for a nucleus, mT = ½ (Z-N) and T = |mT|, ie lowest energy has smallest T
• Example: “isobaric triplet” 14C, 14N, 14O:
14C: Z = 6, N = 8mT = -1, T = 1
14O: Z = 8, N= 6mT = + 1, T = 1
14N mT = 0, T = 0 (g.s) and T = 1 (8 MeV)
ppnn np
Extra T=0states athigher energyin the mT=0nucleus.
4
How do we understand the quantum numbers of nuclei and their excited states?
• integer or half-integer angular momentum depending on whether A is even or odd
• different systematics and energy level spacings for different nuclei
• some nuclei exhibit “single particle” and others “collective” excitations different models to describe this complementary behavior
5
What is the potential energy function V(r) that nuclei are eigenstates of?
This is not an easy question! The N-N interaction (lecture 20) is too complicated tosolve in a many-body system: state-of-the-art can go up to A = 3!
First approximation: a square well potential, width approx. equal to nuclear radius R:
)(rV
r0
oV
B
K
bound state
R
Assume somehow that we can treat the binding of neutrons and protons like electrons in atoms – individual nucleons have wave functions that are eigenstates of some average nuclear potential V(r).
Each nucleon has a binding energy B as shown (E = -B)
Kinetic energy K = (V0 – B)
R 1.2 A1/3 fm; most of the wave function is contained inside the well, so this should be approximately the right nuclear size...
6
Kinetic energy of bound nucleons:
Key point: once we specify the width of the well, the nucleons are confined, and so their kinetic energy is essentially determined by the uncertainty principle:
Simple estimate: Confining box of side 2 fm. px x ~ ħ
2422
22
MeV103~)/(3
/)(
0
xp
xpppp
p
xxxx
x
2422
22
MeV103~)/(3
/)(
0
xp
xpppp
p
xxxx
x
015.02 2
2
M
p
M
K
Conclusion: the motion is non relativistic; K 15 MeV
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Now we can specify the potential parameters:
)(rV
r0
oV
B
K ~ 15 MeV
~ - 3 MeV
R ~ 2 fm
~ - 18 MeV
What next?
• We have a complicated system of A nucleons.
• About half of them are protons, so a repulsive (+ve energy) term has to be added to the square well to account for this (~ few MeV)
How to connect this model to something observable?
Independent particle model:
• Assume independent particle motion in some average nuclear potential V(r) as shown.
• Then we can fill the eigenstates of the potential to maximum occupancy to form a nucleus, as is done with electrons in atoms (to 1st order...)
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Connection to average nuclear properties:
• The binding energy of each nucleon, in our model, is a few MeV.
• The potential energy of a bound nucleon is negative, by ~ 0.3% of its rest mass energy, which therefore has to show up as a decrease in its mass.
• For A nucleons, the total binding energy is:
MmBBA
ii
A
ii
11
MmBBA
ii
A
ii
11
mass of nucleus, M
The average binding energy per nucleon, B/A, can be determined from mass data and used to refine a model for V(r); it ranges systematically from about 1 – 9 MeV as a function of mass number for the stable isotopes.
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Atomic Mass Units:
• By convention, we set the mass of the carbon-12 atom as a standard.
• Denote atomic masses with a “script” M, measured in atomic mass units, U
M (12C) 12.0000000000 ..... U (exact!) 1 U = 931.502 MeV (expt.)
Calculation for carbon-12:
MeV511.0
MeV6.939
MeV2.938
e
n
p
m
m
m
MeV0.178,1121
MeV8.269,116
U
mi
i
MeV8.91)(126 MmCB i
i
Binding energy per nucleon in 12C: B/A = 7.8 MeV;
Contrast to the deuteron 2H: B/A = 1.1 MeV
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The famous Binding Energy per Nucleon curve: (Krane, fig. 3.16)
Most stable: 56Fe,8.8 MeV/ nucleon
gradual decreaseat large A due toCoulomb repulsion
very sharp riseat small A
almost flat, apart from Coulomb effects
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Implications of the B/A curve:
Greater binding energy implies lower mass, greater stability.
Energy is released when configurations of nucleons change to populate the larger B/A region nuclear energy generation, e.g.
Fission reactions at large A release energy because the products have greater binding energy per nucleon than the initial species:
MeV2003nYXnU 133100235 MeV2003nYXnU 133100235
distribution of final products
MeV17.6nHeHH
MeV4.03pH
MeV3.27nHeHH
432
3
322
MeV17.6nHeHH
MeV4.03pH
MeV3.27nHeHH
432
3
322
Binding energyof products is greater than the sum of thebinding energiesof the initial species.
Fusion reactions at small A release substantial energy because the B/A curve rises faster than a straight line at small A:
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A semi-empirical model for nuclear binding energies:
1. Volume and Surface terms:
First consider a 1-dimensional row of nucleons with interaction energy per pair =
total: A
ABA
i
221
ABA
i
221
each has 2 neighbors correctionfor the ends
By analogy, for a 3-d nucleus, there should be both volume and surface terms with the opposite sign, the surface nucleons having less binding energy:
3/13/2 AaaAaAaB SVSV AB 3/13/2 AaaAaAaB SVSV A
B
AAB 2 AA
B 2Approximately constant, with endeffects relatively smaller at large A.
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2. Coulomb term:
R
total charge + Ze
for a uniform sphere,
R
Ze
r
dqrqE
ooCoul 4
)(
4
)( 2
53
This effect increases the total energy and so decreases the binding energy.
Simple model: 3/12 AZaB C
But this is not quite right, because in a sense it includes the Coulomb self energyof a single proton by accounting for the integral from 0 to rp ~ 0.8 fm. The nucleushas fuzzy edges anyway, so we will have to fit the coefficient acto mass data.
3/43/1 )1()1( AZZaAZZaB CC AB 3/43/1 )1()1( AZZaAZZaB CC A
B
Solution: let B scale as the number of proton pairs and include a term:
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3. Symmetry Term:
So far, our formula doesn’t account for the tendency for light nuclei to have Z = N. The nuclear binding energy ultimately results from filling allowed energy levels in a potential well V(r). The most efficient way to fill these levels is with Z = N:
energ
y
0
E
2E
states full belowhere..neutrons protons
Z = N
Simplest model: identical nucleons as a Fermi gas, i.e. noninteracting spin- ½ particles in a box. Two can occupy each energy level. The level spacing ~ 1/A.
A mismatch between Z and N costs an energy price of E at fixed A as shown.
1212 )2()( AZAaANZaB AA1212 )2()( AZAaANZaB AA
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4. Pairing Term:
Finally, recall from slide 1 that for the case of even A, there are 177 stable nuclei with Z and N both even, and only 6 with Z and N both odd. Why?
Configurations for which protons and neutrons separately can form pairs must be much more stable. All the even-even cases have J = 0+, implying that neutrons and protons have lower energy when paired to total angular momentum zero.
Solution: add an empirical pairing term to the binding energy formula:
4/3
1
0
1
AaB ppair
4/3
1
0
1
AaB ppair
with +1 for even-even, 0 for even-odd, and -1 for odd-odd
Full expression:
123/13/2 )2()1(),( AZAaAZZaAaAaAZB ACSV 123/13/2 )2()1(),( AZAaAZZaAaAaAZB ACSV
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Fitting of coefficients to data:
123/13/2 )2()1(),( AZAaAZZaAaAaAZB ACSV 123/13/2 )2()1(),( AZAaAZZaAaAaAZB ACSV
MeV5.15Va
MeV8.16sa
MeV72.0Ca
MeV23Aa
MeV34pa (not shown)
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One more look at the Binding Energy per Nucleon curve:
Most stable: 56Fe,8.8 MeV/ nucleon
gradual decreaseat large A due toCoulomb repulsion
very sharp riseat small A
almost flat, apart from Coulomb effects
Solid line: fit to thesemi-empirical formula
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some largeoscillationsat small mass(next class)