Fundamental Properties

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Fundamental Properties
Mass (kg)
Charge (C) 1.6x10-19 0 -1.6x10-19 Mass (u,
amu) 1.007276 1.008665 0.0005486
1u 931.5 MeV
The nucleus consists of neutrons and protons that
are collectively referred to as Nucleons.
A (NZ)
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Nuclear stability The Strong Force
The nucleus is approximately spherical with a
radius given by r (1.2x10-15) A1/3
m Although the mass of different nuclei
is different, the nuclear density is constant
Why does the nucleus remain intact given the
positive charged protons repel each other with a
very strong repulsive electrostatic force ? Why
dont all nuclei burst apart? Because we know
that nuclei do not (usually) disintegrate, we
must infer there is a force that acts between
neutrons and protons when these particles are
extremely close together.
We call this THE STRONG FORCE. It is very strong
ONLY when the protons and neutrons are VERY CLOSE
to each other (short range force, 10-15 m).
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So, the Strong Force glues together neutrons
and protons within the nucleus. The electrostatic
repulsive force is balanced by the Strong Force.
As the number of protons (ve
charges repelling) increases, the number
of neutrons must increase even more so as to
glue all the nucleons together as a stable
entity. As Z becomes very high (gt 83),
this balance cannot be achieved and the nuclei
are UNSTABLE. In this case, they spontaneously
break apart or disintegrate this process is
called RADIOACTIVITY.
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Mass Defect Binding Energy
Because of the Strong Force, the nucleons inside
the nucleus are held tightly together. Therefore
ENERGY is required to separate the nucleus into
its constituent protons and Neutrons. This energy
is called the BINDING ENERGY.
Einsteins Special Theory of Relativity shows
that mass and energy are equivalent via his
famous equation E mc2. In our case, mass
should really be written as a change in mass, Dm,
which is referred to as a MASS DEFECT. Binding
Energy Dm c2 Where c speed of light
(3.0x108 ms-1)
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Binding Energy of the Helium Nucleus
So for Helium Z 2 and mp 1.672648x10-27
kg N 2 and mn 1.674954x10-27 kg Therefore
Dm 6.6952x10-27 - 6.6447x10-27
0.0505x10-27 kg Binding Energy Dm c2
4.545x10-12 J 28.3 MeV This is the amount of
energy liberated when the nucleons are bound into
a stable helium nucleus. If the process were
reversed and the nucleus split up into discrete,
isolated nucleons, then this is the amount of
energy that would be required PER NUCLEUS!!
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Binding Energy Per Nucleus
This plot shows the binding energy PER NUCLEON,
ie. the binding energy divided by the number of
nucleons in the nucleus. Above A83, this
value begins to decrease. There is less and less
glue per nucleon as the atomic mass increases
(beyond 83) and thus the nuclei become less and
less stable. Furthermore, when light
elements fuse together, there is a release of
energy, but when heavy elements fuse, energy is
required. Similarly, when heavy element
split There is a release of energy.
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Fission
Nuclear fission is the process of splitting
heavy nuclei into two lighter nuclei, described
by the general reaction 10n 23592U
y Y x 10n E In 1934, Enrico
Fermi began a series of experim- ents that
involved bombarding heavy chemical elements with
neutrons in order to create isotopes. He thought
that by hitting uranium with fast neutrons, he
would create an unstable isotope of uranium and
that this isotope would decay by beta emission
to yield a NEW element with atomic number
93. But when he examined the products of the
reaction, he found something different had
occurred.
Fission
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Between 1935 8, Hahn and Strassman of Germany
concluded that the products of the nuclear
reaction appeared to consist of lighter
elements. In reality the reaction
that had occurred was 10n 23592U
14456Ba 8936Kr 3 10n
Energy The only naturally occurring isotope of
uranium in which we can induce fission is 235U
which has an abundance of only 0.72. The
induced fission of this isotope releases an
average of 200 MeV per atom which is equivalent
to 80 million kJ per gram of 235U !! The
attraction of nuclear fission as a power source
can be understood when we compare this figure to
the 50 kJ per gram released when natural gas is
burned.
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The reaction shown earlier is not the only
reaction achieved. Other examples are
10n 23592U 14054Xe 9438Sr
2 10n Energy 10n 23592U
13250Sn 10142Mo 3 10n
Energy In all cases, the masses of the product
nuclei are less than the masses of the Reactants
indicating that energy is released in the
process.
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• Relative to the energy of a spherical liquid
  drop, the liquid-drop energy can be written as

Where the relative surface and Coulomb energies
Bs, and Bc, are functions only of the shape of
the nucleus. The dependence on neutron and proton
numbers is contained in the fissility parameter x
and the surface energy Es (0) of a
spherical Nucleus is given by

where
and fissility parameter becomes
Myers, W. D., Swiatecki, W. J. 1966. Nucl. Phys.
811-60
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Curvature of nuclear surface
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Why still fission?
Despite all of the progress in the understanding
of the fission process, after more than 60 years
of research, still no theory or model is able to
predict all the fission observables ( fission
cross section, post-scission neutron
multiplicities and spectra, fission fragments
properties like mass, charge, total kinetic
energy and angular distributions) in a consistent
way for all possible fissioning systems in a wide
energy range.
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Saddle to scission distance as large as 6 fm

• Liquid Drop Model
• - only the Coulomb (EC) and surface term (ES)
  depend on deformation
• - near g.s. ES increases more rapidly than EC
  decreases V rises
• at larger deformation the situation is reversed
  V drops fission barrier

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I formation of the initial state of the
fissioning nucleus II from the initial state to
scission III from scission to the fission
products formation by prompt processes IV
de-excitation of the fission products by delayed
processes
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• Parameterization of the nuclear shape
• Calculation of the potential energy surface as
  function of deformation
• Determination of the fission path(s), fission
  barrier, transition states.

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potential energy surface potential energy as
function of deformation V(q) fission path
corresponds to the lowest potential energy when
increasing deformation
fission barrier one-dimensional representation
of V as function of one deformation coordinate
(ex.elongation)
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Potential landscape - macroscopic models LDM
gives only a qualitative account of the
phenomenon - microscopic models HFB can not
provide accurate results for Vf yet -
microscopic-macroscopic models LDMshell
correction explains a significant number of
experimental data the most used procedure to
calculate fission barriers
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Strutinskys procedure
V(q)VLD (q)VSh(q)

• The value of the shell correction is , -
  depending on whether the density of
  single-particle states at Fermi surface is great
  or small.
• Negative corrections for actinides
• g.s - permanent deformation

-vicinity of macroscopic saddle point
double-humped fission barrier
-second well
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Variation of the shell correction amplitude
with changing Z,N together with the variation of
the LD potential barrier with changing fissility
parameter (EC/2ES) lead to a variation of the
fission barrier from nucleus to nucleus. - inner
barriers almost constant 5-6 MeV fall rapidly in
Th region - secondary wells depth around 2-3
MeV - outer barriers fall quite strongly from the
lighter actinides (6-7 MeV for Th) to the heavier
actinides (2-3 MeV for Fm).
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Early calculations assumed a maximum
degree of symmetry of the nuclear shape alongthe
fission path and the theoretical predictions were
not in agreement with the experimental barrier
heights and the asymmetric mass distribution of
the fission fragments.
After extensive studies was concluded that axial
asymmetry is indicated for the inner and
reflection asymmetry for the outer barrier.
These results have large implications for the
barrier heights but also for the level densities
at the saddle points.
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Brack, M. et al 1972. Rev. Mod. Phys. 44 326405
109
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The multi- modal fission
(Brosa model)
symmetric super-long (SL) and asymmetric standard
1 (ST I) and II (ST II).
which branch from the standard fission path at
certain bifurcation points on the potential
energy surface.
ST1
ST2
N82
N88
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Symmetric
Asymmetric
Average mass- and TKE values, and the widths of
their distributions are reproduced
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Te10-14 s Limit by LDM Z104
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Fission Cross-section
?a f (E)å ?a f (EJ p) P f (E EJ p) ?a f
(EJ p) cross section of the initial
state formation P f (E EJ p)
fission probability

a - entrance channel E - energy of the incident
particle inducing fission Jp - spin, parity of
the CN state
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Isomeric fission
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Back et al. Phys. Rev. C9, 1924 (1974)
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Formalism for calculation of fission cross-section

• Calculations are sensitive to fission barrier and
  level density at saddle point.
• We have refined the earlier work of Gupta and
  Satpathy (Z. Phys. A326 (1987) 221) based on
  Hugenboltz-Van Hove (HVH) theorem (Physica
  (utrecht)24 (1958) 363 using microscopic-macroscop
  ic approach.
• Brack et al. (Rev. Mod. Phys. 44, (1972) 320)
  finds that changes in surface and Coulomb terms
  are dependent on deformed shapes (c,h)
  parametrization and are similar for all
  actinides at Maxima and Minima of potential
  energy surface.

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• According to HVH theorem
• For asymmetric nuclear matter
• E/A ½ (1ß)en (1- ß)ep
  (1)
• Where E is the total energy, ANZ, en and ep are
  the neutron and proton fermi energies, ß
  (N-Z)/(NZ).
• The eq. (1) is used to arrive at corresponding
  eq. valid for finite nuclei by taking into
  account the finite size effects like surface,
  Coulomb and pairing
• E(N,Z)EF(N,Z) - asA2/3 acZ2 / A1/3d(A,Z)
  (2)
• Where EF is the ground state energy of the
  nucleus with N neutrons and Z protons, as and ac
  are the usual surface and Coulomb coefficients
  and d is pairing term.

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• Using eq. (2) and writing fermi energies in terms
  of binding energies, the mass relation between
  (N,Z), (N-1,Z) and (N,Z-1) can be obtained from
  eq. (1)
• EFN EF(N-1,Z)/(A-1) Z EF(N,Z-1)/(A-1) A
  S(A,Z)/(A-1)
• 
  (3)
• Where defect function S(A,Z) is defined as
• S(A,Z) asA-1/3 - A2/3 (A-1)2/3 ac
  Z2(1-A)A-4/3 ½ (A-1)-1/3
• Z2 (1 ß) (Z-1)2 (1- ß)
  d(A,Z) (1-A-1) ½ (1 ß)
• d(A-1,Z) ½ (1- ß) d(A-1,Z-1)

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• For fission process one can extend eq. (3) for
  deformed nuclei Eq. 2 can be rewritten for
  deformed nuclei
• E(N,Z) EF(N,Z,?) as(?)A2/3 ac(?)Z2 /A1/3 d(?
  ,A,Z) (4)
• Where ? stands for deformation.
• Eq. 3 can be re-written
• EF (N,Z, ?) N EF (N-1, Z, ?)/(A-1) Z
  EF(N,Z-1, ?)/(A-1)-
• A S(A,Z, ?)/(A-1)
  (5)
• If three nuclei have same ground state
  deformation ?o and same barrier deformation, ?
  then we get a relation between the fission
  barrier using eq. (5)
• F(N,Z,?) N F(N-1,Z, ?)/(A-1) Z F(N, Z-1,
  ?)/(A-1)-
• A S(A,Z, ?)/(A-1)
  (6)
• Where F(N,Z, ?) EF(N,Z,?) - EF(N,Z,?o) is the
  fission barrier and
• S(A,Z, ?) S(A,Z, ?)-S(A,Z, ?o) is rather small
  (about 0.15 MeV)

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• The recursion relation can be used to predict the
  fission barrier for any of three nuclei if
  barrier of other two are known provided they have
  same ground state and their barrier deformation.
• This is the case of actinide nuclei as shown by
  Brack et al. (Rev. Mod. Physics. 44, (1972) 320.
  The relevant deformation can be adequately
  described by dimensionless elongation and neck
  coordinates c an h respectively.
• In the present work, we have included pairing
  terms which were not included in the earlier work
  of Gupta and Satpathy and this improves the
  quality of agreement with the experimental data
  on fission barrier given by S. Bjornholm and J.E.
  Lynn (Rev. Mod. Phys. 52 (1980) 725).

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• We use same analytical formula for both the
  barriers, however the coefficients are different
  in two cases.
• The new refined analytical formula for the
  fission barrier Via/b of a nucleus (A,Z,N) is
• Vi as Bsi A2/3 ac Bci Z2/3 pZ?N k ½
  1(-1)Z dZ
• ½ 1(-1)N dN
• The first two terms are the macroscopic terms of
  the formula.
• For these we adopt values from Gupta and Satpathy
  (Z. Phys. A326 (1987) 221) and use as
  19.1-2.84((N-Z)/A) 2 MeV and
• Ac 0.72 MeV. The next three terms are
  microscopic terms and satisfy Hugenholtz and Van
  Hove theorem. The last two terms denote the
  pairing terms.

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• The changes of surface and Coulomb terms Bsi and
  Bci are taken from Brack et al. (Rev. Mod. Phys.
  44, (1972) 320) their values are listed in Table
  I.
• The associated five parameters p, ?, k, dZ and dN
  are determined by least square fits to the
  fission barrier data by Bjornholm and Lynn (Rev.
  Mod. Phys. 52 (1980) 725). For Z89 to Z98.
• The deduced parameters and ?2 / degree of freedom
  are also listed in Table I.
• The fits are shown in Fig.1 and Fig. 2

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Fig.1 Inner barrier Va experimental and
calculated with the barrier formula
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Fig.2 Outer barrier Vb experimental and
calculated with the barrier formula
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CALCULATIONS AND RESULTS

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EMPIRE-specific level densities Features Collect
ive enhancements due to nuclear vibration and
rotation. Super-fluid model below critical
excitation energy (GSF) Fermi gas model above
critical excitation energy (FG) Rotation induced
deformation (spin dependent) gt moments of inertia
ATILNO multiplies a-parameter (the
only possible intervention)
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dW being the shell correction ã is
asymptotic value and includes effects of
deformation.
EMPIRE-specific energy dependent following
Ignatyuk et al.
Shell effects wash out with increasing energy
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• One can see that the nucleus 233Pa (which
  corresponds to 239Np,T1/22.3 days in the usual
  U-Pu cycle) has sufficiently long half life
  (T1/227 days ) for capturing neutrons and can
  therefore seriously perturb the production of
  233U production.
• The build-up and decay of 233Pa
  influences the breeding and the reactivity
  behaviour and it is referred to as the
  Protactinium effect.

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• Tovesson et al. have carried out a number of
  studies
• and measured neutron induced fission
  cross-section using
• radioactive target of 233Pa between 1.0 to
  3.0 MeV (PRL 88
• (2000) 06502). Recent studies extended the
  energy range to
• 8.5 MeV (Nucl. Phys. A733, (2004) 3)

• Recently Mesa et al. (Phys. Rev. C68
  (2003) 054608)
• carried out a new calculation for this
  cross-section. Their
• calculation is tedious and calculate the
  fission barriers
• using the potential energy surface with
  shape
• parametrization and the level density at
  the barrier
• maxima.

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(RFRM)
Macroscopic Model of Rotating Nuclei
A.J. Sierk, Phys. Rev. C33, 2039(1986)

• If L is angular momentum in units ?, The
  rotational
• energy is given by

where
is moment of inertia of sphere
y0.057
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Rotational energy for an isolated nucleus
Where Moment of Inertia
non rotating case
248Cf BfLDM 4.0MeV BfFRM 2.0 MeV
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(Tri-axial)
(Bf0)
Saddle shapes at L0
L has same effect as fissility,ie. with
increasing L, elongation decreases and neck
increases for saddle shapes
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Fission barrier and moment of inertia at saddle
point are Calculated using BARFIT and MOMFIT
subroutines using Global FITS