Nuclear Reactions

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Nuclear Reactions
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Nuclear Reactions

• Binding Energies
• The mass law below represents the masses of
  thousands of nuclei with a few parameters
• B(Z(mpme)(A-Z)mn - M(A,Z))c2
• Mass Excess ?M 9.31.478MeV (M(A,Z)-A) M in AMU
• Q value - energy released in exit channel of rxn
  assuming incoming kinetic energy small
  ??Min - ??Mout
• B/A binding energy per nucleon

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Nuclear Reactions

• Mass terms
• M(A,Z) Zmp (A-Z)mn
• m1 -a1A volume term
• m2 a2A2/3 surface tension
• m3 a3(A/Z - Z)2/A symmetry term from Fermi
  energy of pn Fermi-Dirac gases
• m4 a4 Z2/A1/3 Coulomb repulsion of protons
• m5 ?(A) pairing energy - paired p or n more
  tightly bound
• set to find minimum in mass for
  a given A - valley of stability

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Nuclear Reactions

• valley of stability - At high Z, nuclei are
  stable only if neutron gt proton - coulomb
  term otherwise too large
• High Z elements neutron rich - initla stellar
  composition n poor - need rxns which are n sources

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Nuclear Reactions

• The Coulomb barrier
• Classical limit
• Rnucleus r0A1/3 r01.2x10-13cm
• r gtgt ? h/mc x c/v
• QM limit
• ?compton h/mc 1.13x10-13cm
• for v/c 0.25, ? 4.5x10-13cm
• Rxn rate for flux of particles Npv into a target
  of area a, thickness x, and density Nt

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Nuclear Reactions

• In center of mass frame

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Nuclear Reactions

• Assuming Boltzmann dist.

Integrand max when ?E/kTb/vE is a minimum gives
shape of nuclear potential
Coulomb part of potential
?v2/2
nuc. pot.
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Nuclear Reactions

• Resonances
• After capture the new particle may be in an
  excited state of the compound nucleus. This
  increases the cross section for capture in a
  narrow energy range around the excited state with
  width ?E /?state
• Network equations

A term exists for every possible rxn channel
which creates or destroys j finite difference
approx
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Nuclear Reactions

• Terms such as Yj(t?t)Yk(t?t) go to Yj(t)?k
  Yk(t)?jYj(t)Yk(t)
• linearize - discard higher order terms in ?
• An eqn linear in unknowns ? can be written for
  each species
• The eqn for each species j contains a term ?k for
  each species k connected to j by a rxn
• Write as a matrix ?AB where ? is a column of ?s
• A is a JxK matrix for J species with K terms -
  generally JK with most entries 0
• B is a column of RHS rxn coefficients YaYb?NAlt?vgt
• Want ?s ?Y to determine change in Y
• Solve ?BA-1
• This formulation automatically includes reverse
  rates for rxns since for every matrix element j,k
  there is an element k,j which describes reverse
  rxn

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Nuclear Reactions

• Nuclear rxns in stars can progress down three
  paths
• Complete burning - most familiar H?He, He?CO ash
  is a minimum energy state
• Steady state - dYi/dt0 from contributions of
  several channels - CNO in H burning reach steady
  state abundances for a given T,?
• Equilibrium - forward/reverse rates balance. Get
  broad distribution of abundances determined by
  chemical potentials - minimize thermodynamic free
  energy of system
• Limiting rates determine speed of reaction -
  often weak interactions e.g. in PP chain
  1H(p,?)2D ?109yr

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The Asymptotic Giant Branch

• When He core exhausted He shell burning begins
• Like H shell burning He shell drives the star
  redward - moves star along the Asymptotic Giant
  Branch roughly parallel to but higher in
  luminosity than the RGB

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The Asymptotic Giant Branch

• When He core exhausted He shell burning begins
• Like H shell burning He shell drives the star
  redward - moves star along the Asymptotic Giant
  Branch roughly parallel to but higher in
  luminosity than the RGB
• Second dredge-up brings H burning products to
  surface
• H shell quenched until He shell moves out far
  enough to heat shell to burning T
• of stars on AGB/ of stars on HB gives
  constraint on amount of time star spends in core
  He burning

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The Asymptotic Giant Branch

• Extreme density gradients outside degenerate
  corre and burning shells

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The Asymptotic Giant Branch

• Center of star is degenerate and cooling from
  weak ? emission - peak T not in core

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The Asymptotic Giant Branch

• Star has extremely compact core - most of radius
  is extended envelope

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The Asymptotic Giant Branch

• Star has extremely compact core - most of radius
  is extended envelope

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The Asymptotic Giant Branch

• Double shell burning or Thermal Pulse AGB
• q(He) 0.1q(H) so He shell catches up to H shell
• As He shell approaches H shell material expands,
  H shell quenched
• He burns outward, runs out of fuel, quenched H
  shell restarts, eats outward, ash builds up
• He shell ignites, repeat

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The Asymptotic Giant Branch

• Double shell burning or Thermal Pulse AGB
• During He shell phases envelope convection
  penetrates deeply into star
• He shell produces small convective shell
• Non-convective mixing allows transport between
  shells
• mixing 12C into H flame zone or p into He flame
  gives 12C(p,?)13N(?decay)13C
• 13C(?,n)16O is a neutron source - only works when
  p and He burning can mix

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The Asymptotic Giant Branch

• Double shell burning or Thermal Pulse AGB
• s-process - slow n capture onto Fe peak seed
  nuclei - each n captured has time to ? decay to a
  proton, increasing Z
• s-process takes place in intershell region where
  n produced primarily in intermediate mass stars
  just above maximum mass for He flash
• Produces species with Agt90
• 3rd dredge-up (actually numerous dredge-ups for
  each thermal pulse cycle) brings partial He
  burning products to surface with s-process
  enhancements - most efficient at low metallicity
  - C stars, Sr stars

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The Asymptotic Giant Branch

• Double shell burning or Thermal Pulse AGB
• Produces species with Agt90

s-process peaks where p n form closed shells -
p and n magic numbers i.e. 208Pb with Z82,
N126, both magic numbers even Z and even A
nuclei more abundant
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AGB Mass Loss
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AGB Mass Loss

• Often highly asymmetric (bipolar)
• AGB stars generally very cool - spectra dominated
  by molecular species
• H2O, TiO, VO, Sr, Ba compounds, Si2O3 SiC, C2,
  Buckyballs in carbon stars
• Complex molecular spectra and low T allow line
  blanketing - much of high L goes into
  accelerating wind
• Atmospheres of cool stars dust rich
• winds from direct radiation pressure
• dust formation region can act like ? mechanism -
  drive pulsations which become non-linear and
  create shocks in low ? stellar atmosphere

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AGB Mass Loss

• Thermal pulses during double shell burning can
  drive mass loss episodes
• Shell flashes - if H or He shell is degenerate
  when it ignites small explosion drives mass loss,
  may revivify proto-WD as red giant (Sakurais
  object)
• Small envelope above a burning shell can be
  removed in a short event - planetary nebula
• Fast wind from proto-WD evacuates bubble, causes
  Rayleigh-Taylor instabilities in swept-up shell
• add ionizing radiation from central star and get
  planetary nebula
• Low mass stars have only compact ionized bubble,
  high mass disperse envelope very quickly - only
  intermediate masses have visible PN with lifetime
  10,000yr

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Morphology of Planetary Nebulae

• Many PN/proto-PN strongly bipolar
• IR and polarization show thick dusty torus
• Some axisymmetry due to rotation
• Mechanism for tight collimation unknown - B
  fields or companions possible
• Fliers - Fast, low ionization emission regions -
  clumps moving at hundreds of km s-1 near symmetry
  axis - mechanism unknown

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Morphology of Planetary Nebulae

• Clumping - Shell of swept-up material breaks into
  dense clumps (n104-6cm-3)
• Two possible mechanisms - Rayleigh-Taylor
  instability from fast, low density wind impacting
  shell
• Or thermal instabilities - rapid efficient
  cooling ahead of shock causes fragmentation on
  scale where sound crossing time cooling time

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