III. Nuclear Physics that determines the properties of the Universe

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III. Nuclear Physics that determines the
properties of the Universe
Part I Nuclear Masses
1. Why are masses important ?
1. Energy generation
nuclear reaction A B C
if mAmB gt mC then energy Q(mAmB-mC)c2 is
generated by reaction
Q-value Q Energy generated (gt0) or consumed
(lt0) by reaction
2. Stability
with Qgt0 ( or mAgt mB mC)
if there is a reaction A B C
then decay of nucleus A is energetically possible.
nucleus A might then not exist (at least not for
a very long time)
3. Equilibria
for a nuclear reaction in equilibrium abundances
scale with e-Q
(Saha equation)
Masses become the dominant factor in determining
the outcome of nucleosynthesis
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2. Nucleons
Mass Spin Charge
Proton 938.272 MeV/c2 1/2 1e
Neutron 939.565 MeV/c2 1/2 0
size 1 fm
3. Nuclei
nucleons attract each other via the strong force
( range 1 fm)
a bunch of nucleons bound together create a
potential for an additional
neutron
proton(or any other charged particle)
V
V
Coulomb Barrier Vc
R 1.3 x A1/3 fm
Potential
Potential
R
R
r
r
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4. Nuclear Masses and Binding Energy
Energy that is released when a nucleus is
assembled from neutrons and protons
mp proton mass, mn neutron mass, m(Z,N)
mass of nucleus with Z,N

• Bgt0
• With B the mass of the nucleus is determined.
• B is roughly A

Masses are usually tabulated as atomic masses
m mnuc Z me Be
Nuclear Mass 1 GeV/A
Electron Binding Energy13.6 eV (H)to 116 keV
(K-shell U) / Z
Electron Mass 511 keV/Z
Most tables give atomic mass excess D in MeV
(so for 12C D0)
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Can be understood in liquid drop mass model
(Weizaecker Formula) (assumes incompressible
fluid (volume A) and sharp surface)
Volume Term
Surface Term surface area (Surface nucleons
less bound)
Coulomb term. Coulomb repulsion leads to
reduction uniformly charged sphere has E3/5
Q2/R
Asymmetry term Pauli principle to protons
symmetric filling of p,n potential boxes has
lowest energy (ignore Coulomb)
lower totalenergy more bound
neutrons
protons
protons
neutrons
x 1 ee x 0 oe/eo x (-1) oo
Pairing term even number of like nucleons
favoured
(eeven, oodd referring to Z, N respectively)
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Best fit values (from A.H. Wapstra, Handbuch der
Physik 38 (1958) 1)
aV aS aC aA aP
15.85 18.34 0.71 92.86 11.46
in MeV/c2
Deviation (in MeV) to experimental masses
(Bertulani Schechter)
something is missing !
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Shell model(single nucleon energy levels)
are not evenly spaced
shell gaps
less boundthan average
more boundthan average
need to addshell correction termS(Z,N)
Magic numbers
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Understanding the B/A curve
neglect asymmetry term (assume reasonable
asymmetry) neglect pairing and shell correction -
want to understand average behaviour
then
constas strong force hasshort range
Coulomb repulsion has long range- the more
protons the more repulsion favours small (low Z)
nuclei
surface/volume ratiofavours large nuclei
maximum around Fe
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5. Decay - energetics and decay law
Decay of A in B and C is possible if reaction A
BC has positive Q-value
BUT there might be a barrier that prolongs the
lifetime
Decay is described by quantum mechanics and is a
pure random process, with a constant probability
for the decay to happen in a given time interval.
N Number of nuclei A (Parent) l decay rate
(decays per second and parent nucleus)
therefore
lifetime t1/l
half-life T1/2 t ln2 ln2/l is time for half
of the nuclei present to decay
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6. Decay modes
for anything other than a neutron (or a neutrino)
emitted from the nucleusthere is a Coulomb
barrier
V
Coulomb Barrier Vc
unboundparticle
R
r
Potential
If that barrier delays the decay beyond the
lifetime of the universe ( 14 Gyr)we consider
the nucleus as being stable.
Example for 197Au -gt 58Fe 139I has Q 100
MeV ! yet, gold is stable.
not all decays that are energetically possible
happen
most common

• b decay
• n decay
• p decay
• a decay
• fission

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(rates later )
6.1. b decay
p
n
conversion within a nucleus via weak interaction
Modes (for a proton/neutron in a nucleus)
b decay
electron capture
b- decay
Electron capture (or EC) of atomic electrons or,
in astrophysics, of electrons in the surrounding
plasma
Q-values for decay of nucleus (Z,N)
with nuclear masses
with atomic masses
Qb / c2 mnuc(Z,N) - mnuc(Z-1,N1) - me
m(Z,N) - m(Z-1,N1) - 2me
QEC / c2 mnuc(Z,N) - mnuc(Z-1,N1) me
m(Z,N) - m(Z-1,N1)
Qb- / c2 mnuc(Z,N) - mnuc(Z1,N-1) - me
m(Z,N) - m(Z1,N-1)
Note QEC gt Qb
by 1.022 MeV
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Q-values with D values
Note
Q-values for reactions that conserve the number
of nucleons can also be calculated directly
using the tabulated D values instead of the
masses
Example 14C -gt 15N e ne
(for atomic Ds)
Q-values with binding energies B
Q-values for reactions that conserve proton
number and neutron number can be calculated
using -B instead of the masses
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b decay basically no barrier -gt if energetically
possible it usually happens
(except if another decay mode dominates)
therefore any nucleus with a given mass number A
will be converted into the most
stable proton/neutron combination with mass
number A by b decays
(Bertulani Schechter)
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Typical part of the chart of nuclides
red proton excessundergo b decay
blue neutron excessundergo b- decay
Z
N
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Typical b decay half-lives
very near stability occasionally Mios of
years or longer
more common within a few nuclei of stability
minutes - days
milliseconds
most exotic nuclei that can be formed
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6.2. Neutron decay
When adding neutrons to a nucleus eventually the
gain in binding energy dueto the Volume term is
exceeded by the loss due to the growing asymmetry
term
then no more neutrons can be bound, the neutron
drip line is reached
beyond the neutron drip line, neutron decay
occurs
(Z,N) (Z,N-1) n
Q-value
Qn m(Z,N) - m(Z,N-1) - mn
(same for atomic and nuclear masses !)
Neutron Separation Energy Sn
Sn(Z,N) m(Z,N-1) mn - m(Z,N) -Qn for
n-decay
Neutron drip line
Sn 0
beyond the drip line
Snlt0
the nuclei are neutron unbound
As there is no Coulombbarrier, and n-decay is
governed by the strong force,for our purposes
the decay is immediate and dominates all other
possible decay modes
Neutron drip line very closely resembles the
border of nuclear existence !
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Example Neutron Separation Energies for Z40
(Zirconium)
add 37 neutrons
valley of stability
neutron drip
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6.3. Proton decay
same for protons
Proton Separation Energy Sp
Sp(Z,N) m(Z-1,N) mp - m(Z,N)
Proton drip line
Sp 0
beyond the drip line
Snlt0
the nuclei are neutron unbound
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N40 isotonic chain
add 7protons
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Main difference to neutron drip line

• When adding protons, asymmetry AND Coulomb term
  reduce the binding therefore steeper drop of
  proton separation energy - drip line reached much
  sooner
• Coulomb barrier (and Angular momentum barrier)
  can stabilize decay, especially for higher Z
  nuclei (lets say gt Z50)

Nuclei beyond (not too far beyond) can therefore
have other decay modes than p-decay. One has
to go several steps beyond the proton drip line
before nuclei cease to exist (how far depends
on absolute value of Z).
Note have to go beyond Z50 to prolong
half-lives of proton emitters so that they
can be studied in the laboratory
(microseconds)
for Zlt50 nuclei beyond the drip line are very
fast proton emitters (if they exist at all)
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Example Proton separation energies of Lu (Z71)
isotopes
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Pb (82)
Sn (50)
proton dripline
Fe (26)
neutron dripline
note odd-even effect in drip line !(p-drip even
Z more bound - can take away more ns) (n-drip
even N more bound - can take away more ps)
protons
H(1)
neutrons
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6.4. a decay
emission of an a particle ( 4He nucleus)
Coulomb barrier twice as high as for p emission,
but exceptionally strong bound, so larger Q-value

• emission of other nuclei does not play a role
  (but see fission !) because of
• increased Coulomb barrier
• reduced cluster probability

Q-value for a decay
lt0, but closer to 0 with larger A,Z
large A therefore favored
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lightest a emitter 144Nd (Z60)
(Qa1.9 MeV but still T1/22.3 x 1015 yr)
beyond Bi a emission ends the valley of stability
!
yelloware a emitter
the higher the Q-value the easier the Coulomb
barrier can be overcome(Penetrability
)and the shorter the
a-decay half-lives
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6.5. Fission
Very heavy nuclei can fission into two parts
(Qgt0 if heavier than iron already)
For large nuclei surface energy less important -
large deformations less prohibitive. Then, with
a small amount of additional energy (Fission
barrier) nucleuscan be deformed sufficiently so
that coulomb repulsion wins over nucleon-nucleon
attraction and nucleus fissions.
Separation
(from Meyer-Kuckuk, Kernphysik)
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Real fission barriers
Fission barrier depends on how shape is changed
(obviously, for example. it is favourable to
form a neck).
Real theories have many more shape parameters -
the fission barrier is then a landscape with
mountains and valleys in this parameter space.
The minimum energy needed for fission along the
optimum valley is the fission barrier
Example for parametrization in Moller et al.
Nature 409 (2001) 485
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Fission fragments
Naively splitting in half favourable (symmetric
fission)
There is a asymmetric fission mode due to shell
effects
(somewhat larger or smaller fragment than exact
half might be favouredif more bound due to magic
neutron or proton number)
Both modes occur
Example from Moller et al. Nature 409 (2001) 485
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If fission barrier is low enough spontaneous
fission can occur as a decay mode
green spontaneous fission
spontaneous fission is the limit of existence
for heavy nuclei
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Understanding some solar abundance features from
nuclear masses
1. Nuclei found in nature are the stable ones
(right balance of protons and neutron, and
not too heavy to undergo alpha decay or
fission) 2. There are many more unstable nuclei
that can exist for short times after their
creation (everything between the proton and
neutron drip lines and below spont. Fission
line) 3. Alpha particle is favored building block
because

• comparably low Coulomb barrier when fused with
  other nuclei
• high binding energy per nucleon - reactions that
  use alphas have larger Q-values

explains peaks at nuclei composed of multiples of
alpha particles in solar system abundance
distribution
4. Binding energy per nucleon has a maximum in
the Iron Nickel region
in equilibrium these nuclei would be favored
iron peak in solar abundance distribution
indicates that some fraction of matter was
brought into equilibrium (supernovae !)
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Surprises
1. ALL stable nuclei (and even all that we know
that have half-lives of the order of the age
of the universe, such as 238U or 232Th) are found
in nature
Explanation ?
2. Nucleons have maximum binding in 62Ni. But
the universe is not just made out of 62Ni !
Explanation ?