Folie 1

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Reaction Rates
Marialuisa Aliotta
School of Physics University of Edinburgh

• principles of stellar structure and evolution
• general features of thermonuclear reactions
• experimental approach

Second European Summer School on Experimental
Nuclear Astrophysics St. Tecla, Sept. 28th Oct.
5th 2003
2
The Macro-cosmos some observables
Luminosity vs. surface temperature
Hertzsprung-Russel (HR) Diagram
Surprise!
SUPERGIANTS
No chaos, but order!
GIANTS
Stefans law L 4?R2?T4
sun
Luminosity
WHITE DWARFS
MAIN SEQUENCE
? Temperature K

• 95 of all stars in MAIN SEQUENCE
• highest probability of observing them in this
  stage
• (cf. adulthood for human beings)

Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
3
The Macro-cosmos some observables
Mass-Luminosity relationship (for main sequence
stars only)
L M4 ? more massive stars evolve more rapidly
mass (Msun) lifetime (years) 1 1010
5 108 10 107
L, T, M cannot take up ANY values
ORDER!
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
4
The Macro-cosmos some observables
Abundance curve of the elements
Data sources Earth, Moon, meteorites,
stellar (Sun) spectra, cosmic rays...

• Features
• distribution everywhere similar
• 12 orders-of-magnitude span
• H 75
• He 23
• C ? U 2 (metals)
• D, Li, Be, B under-abundant
• exponential decrease up to Fe
• peak near Fe
• almost flat distribution beyond Fe

Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
5
Experimental nuclear astrophysics
(EXPERIMENTAL) NUCLEAR ASTROPHYSICS

• study energy generation processes in stars
• study nucleosynthesis of the elements

• What is the origin of the elements?
• How do stars and galaxies form and evolve?
• What powers the stars?
• How old is the universe?

MACRO-COSMOS intimately related to MICRO-COSMOS
NUCLEAR PHYSICS KEY for understanding
Courtesy M. Arnould
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
6
Quiescent stages of stellar evolution
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
7
Principles of stellar structure and evolution
quiescent evolution

• Stellar structure and evolution controlled by
• Gravity ? collapse
• Internal pressure ? expansion

Star composed of many particles (1057 in the
Sun) Total energy a) mutual gravitational
energy of particles (?) b) internal
(kinetic) energy of particles (including photons)
(U)
For an ideal gas in hydrostatic equilibrium
2U ? 0 virial
theorem
Assume pressure imbalance
gravitational contraction sets in
amount of energy released -??
internal energy change to restore equilibrium
?U - ½ ??
gas temperature increases
energy excess - ½ ?? lost from star in form of
radiation
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
8
Principles of stellar structure and evolution
quiescent evolution
gravitational contraction of gas (mainly H)
increase of central temperature
if T high enough nuclear
burning takes place
HYDROGEN BURNING (1st equilibrium)
4H ? 4He 2b 2n 26 MeV
ash of nuclear burning
energy source
gravitational collapse is halted ? star
undergoes phase of hydrostatic equilibrium
MAIN SEQUENCE STARS
Here T 10 15 X106 K and r 102 gcm-3
are required
M gt 0.1 M?
(Jupiter failed star)
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
9
Principles of stellar structure and evolution
quiescent evolution
Hydrogen burning
Two main mechanisms proton-proton chain
and CNO cycle
Energy production rate
M lt 1.5 M? ? T6 lt 30
? p-p chain
M ? 1.5 M? ? T6 gt 30
? CNO cycle (also depends on CNO abundance)
(Fiorentinis lecture)
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
10
Principles of stellar structure and evolution
quiescent evolution
H exhausted in core isothermal He
core contraction sets in
H-burning shell
temperature increases
R 10-100 Ri ? Ts 3-4x103 K
Wiens law ?maxT const.
RED GIANT STARS
contracting core expanding envelope
when T 108 K and r 103 gcm-3 (minimum
mass 0.5 M?)
HELIUM BURNING (2nd equilibrium)
3a ? 12C 12C(a,g)16O
8 MeV
energy source
nuclear burning ashes
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
11
Principles of stellar structure and evolution
quiescent evolution
12C/16O BURNING
12C ashes Ne, Na, Mg 16O ashes Al, Si
major ash 28Si
SUPER RED-GIANT STARS
A 40-65
28Si MELTING
PRE-SUPERNOVA STARS
major ash 56Fe
further reactions become endothermic final
gravitational collapse SUPERNOVA EXPLOSION
(type II)
T, r
M ? 8 M?
remnant neutron star or black hole
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
12
Principles of stellar structure and evolution
summary
Evolution stages of a 25 M? star
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
13
Principles of stellar structure and evolution
quiescent evolution
Main parameters 1) initial mass ( ? central
temperature) 2) initial chemical composition (
? nuclear processes)

• Energy generation rate
• ? Tn
• n 4 (H-burning)
• n 30 (C-burning)

innermost regions only contribute to nuclear
burning
e.g. 1/10 M for H-burning less for subsequent
stages
H-burning ? MAIN SEQUENCE longest stage of stars
lifetime
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
14
Explosive stages of stellar evolution
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
15
Principles of stellar structure and evolution
explosive evolution
NOVAE
sudden increase in stars luminosity (L
104 106 Li and t 1 h 1 d)
semi-detached binary system
White Dwarf less evolved star (e.g. Red Giant)
H-rich mass transfer from RG to WD
degenerate matter ? P and T uncoupled
thermonuclear runaway ? cataclysmic explosion
temperature and density increase on WDs surface
(p,?) and (?,p) reactions on proton-rich nuclei
T gt 108 K ? gt 103 g cm-3
determine nature of nova phenomenon
nucleosynthesis up to A 60 mass region
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
16
Principles of stellar structure and evolution
explosive evolution
X-RAY BURSTERS X-RAY PULSARS
semi-detached binary system
Neutron star less evolved star
T 109 K ? 106 g cm-3
(?,p) and (p,?) reactions on proton-rich nuclei
nucleosynthesis up to A 80-100 mass region
intense X-ray fluxes
CORE-COLLAPSE SUPERNOVAE
end stage of M 8-30 M? stars
core collapse rebound shock wave outer
layers blown off Neutron Star or Black Hole
remnants
T gt 109 K ?n gt 1020 g cm-3 seed nuclei in Fe
region
(n,?) reactions on neutron-rich nuclei followed
by ? decays
nucleosynthesis of n-rich elements through
r-process
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
17
Stellar life cycle
BIRTH gravitational contraction
Interstellar gas
Stars
mixing of interstellar gas
thermonuclear reactions

• energy production
• stability against collapse
• synthesis of metals

abundance distribution
explosion DEATH
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
18
Thermonuclear reactions in stars
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
19
Thermonuclear reactions in stars properties
of nuclei
Aston measurements of atomic masses
?E ?Mnc2
Mnucl lt ?mp ?mn
enormous energy stored in nuclei!
Rutherford (1919) discovery of nuclear
reactions

• liberate nuclear energy source
• complex nuclides formed through reactions

Q (m1m2)-(m3m4)c2 gt 0
amount of energy liberated in nuclear reaction
Binding energy curve
spontaneous nuclear processes
Q gt 0
fusion up to Fe region fission of heavy nuclei
H most abundant element in the Universe
FUSION reactions most effective in stars
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
20
Thermonuclear reactions in stars
general features definitions
Consider reaction 1 2 ? 3 4
Q12 gt 0
( ? known from atomic mass tables)
Reaction cross section ? ? probability for a
reaction to occur
Dimension area Unit barn (b) 10-24 cm2
In general not possible to determine
reaction cross section from first principles

• cross sections depend on nature of force
  involved
• cross sections are energy (i.e. velocity)
  dependent

However
r
Reaction rate
v?(v)
N1N2
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
21
Thermonuclear reactions in stars
general features definitions
In stellar plasma velocity of particles
varies over wide range
Quiescent stellar burning non-relativistic,
non-degenerate gas in thermodynamic equilibrium
at temperature T
Maxwell-Boltzmann distribution
?(v) ? exp exp
Probability ?(E)
?(E) ? exp(-E/kT)

• reduced mass
• v relative velocity

?(E) ? E
Energy
kT
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
22
Thermonuclear reactions in stars
general features definitions
Energy production rate ?12
R12 Q12
Mean lifetime of nuclei X against destruction by
nuclei a
energy production as star evolves
change in abundance of nuclei X
lt?vgt KEY quantity
to be determined from experiments and/or
theoretical considerations
as star evolves, T changes ? evaluate ltsvgt
for each temperature
NEED ANAYLITICAL EXPRESSION FOR ?!
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
23
Thermonuclear reactions in stars
reaction mechanisms
Consider reaction a X ? b Y
(b particle or photon)
Non-resonant process
One-step process leading to final nucleus Y
? ? ltbY lHl aXgt2
single matrix element

• occurs at all interaction energies
• cross section has WEAK energy dependence

Resonant process
Two-step process 1) compound nucleus formation
a X ? C 2) decay of compound nucleus
C ? b Y
? ? ltbY lHl Cgt2 ltC lHl aXgt2
two matrix elements

• occurs at specific energies
• cross section has STRONG energy dependence

Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
24
Reactions between charged particles
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
25
Thermonuclear reactions in stars
charged particles
charged particles
Coulomb barrier
V
energy available from thermal motion
Coulomb potential
Ekin kT (keV)
Ecoul Z1Z2 (MeV)
kT 8.6 x 10-8 TK keV
tunnel effect
r
r0
T 15x106 K (e.g. our Sun) ? kT 1 keV
T 1010 K (Big Bang) ? kT 2 MeV
nuclear well
reactions occur through TUNNEL EFFECT
tunneling probability P ? exp(-2??)
during quiescent burnings kT ltlt Ec
2ph GAMOW factor
determines exponential drop in abundance curve!
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
26
Thermonuclear reactions in stars
non-resonant reactions
Non-resonant reactions
geometrical factor (particles de Broglie
wavelength)
interaction matrix element
penetrability probabilitydepends on
projectilesangular momentum ? and energy E
? (E) exp(-2??) S(E)
(for s-waves only!)
non-nuclear origin STRONG energy dependence
nuclear origin WEAK energy dependence
Above relation defines ASTROPHYSICAL
S(E)-FACTOR
N.B.
If angular momentum is non zero ?
centrifugal barrier must also be
taken into account
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
27
Thermonuclear reactions in stars
Gamow peak
With above definition of cross section
lt?vgt12
S(E) exp dE
f(E)
varies smoothly with energy
governs energy dependence
MAXIMUM reaction rate
?E0 lt E0
only small energy range contributes to reaction
rate ? OK to set S(E) S(E0) const.
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
28
Thermonuclear reactions in stars
Gamow peak
Gamow peak
most effective energy region for thermonuclear
reactions
energy window of astrophysical interest
E0 f(Z1, Z2, T)
varies depending on reaction and/or temperature
Examples T 15x106 K (T6 15)
area of Gamow peak (height x width) lt?vgt
STRONG sensitivity to Coulomb barrier
separate stages H-burning He-burning C/O-bur
ning
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
29
Thermonuclear reactions in stars
resonant reactions
Resonant reactions
Breit-Wigner formula
insert in expression for reaction rate,
integrate and get
lt?vgt12
exp
(for single resonance)
resonance strength (integrated cross section over
resonant region)
low-energy resonances (ER ? kT) dominate reaction
rate
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
30
Thermonuclear reactions in stars
resonant reactions
Breit-Wigner formula energy dependence of
partial ?a(E), ?b(E) and total ?(E) widths
N.B. Overlapping broad resonances of same Jp ?
interference effects
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
31
Thermonuclear reactions in stars
resonant reactions
3. Sub-threshold resonances
any exited state has a finite width
? h/?
high energy wing can extend above particle
threshold
cross section can be entirely dominated by
contribution of sub-threshold state(s)
Example 12C(?,?)16O (Gialanellas lecture
Schürmanns talk)
TOTAL REACTION RATE
??v?tot ??v?r ??v?nr
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
32
Reactions with neutrons
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
33
Thermonuclear reactions in stars
neutron captures
NO Coulomb barrier
neutrons produced in stars are quickly thermalised
E0 kT relevant energy (e.g. T 1-6x108
K ? E0 30 keV)
accounts for almost flat abundance distribution
beyond iron peak
ltsvgt const ltsTvTgt
Typically
neutron-capture cross sections can be measured
DIRECTLY at relevant energies
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
34
Experimental approach laboratory requirements
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
35
Experimental approach
general features
Quiescent burning stages of stellar evolution
T 106 - 108 K ? E0 100 keV ltlt Ecoul
? tunnel effect ? 10-18 barn lt ? lt 10-9
barn ? average interaction time ? lt?vgt-1
109 y unstable species DO NOT play
significant role
FEATURES
10-18 b lt ? lt 10-9 b ? poor signal-to-noise
ratio ? major experimental challenge ?
extrapolation procedure required
PROBLEMS
poor signal-to-noise ratio ? long
measurements ? ultra pure targets ? high
beam intensities ? high detection efficiency
REQUIREMENTS
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
36
Experimental approach
extrapolation
Experimental procedure
measure ?(E) over as wide a range as possible,
then EXTRAPOLATE down to Gamow energy region
around E0!
CROSS SECTION
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
37
Thermonuclear reactions in stars
non resonant reactions
Example
cross section
S-factor
Data EXTRAPOLATION down to astrophysical energies
REQUIRED!
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
38
Experimental approach
extrapolation
S(E)-FACTOR
S(E)
extrapolation
direct measurement
LINEAR SCALE
non resonant process
sub-threshold resonance
interaction energy E
0
-Er
DANGER OF EXTRAPOLATION !
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
39
Experimental approach
extrapolation
ALTERNATIVE SOLUTIONS

• Go UNDERGROUND ? reduce (cosmic) background
• example LUNA facility ? (Junkers
  lecture)
• Use INDIRECT methods ? (Figueras lecture)

INTRINSIC LIMITATION
At lower and lower energies ELECTRON SCREENING
EFFECT sets in
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
40
The electron screening
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
41
Experimental approach
electron screening
penetration through Coulomb barrier between BARE
nuclei

• in stellar plasmas ions in sea of free
  electrons

Ec
bare
Coulomb potential
Debye-Hückel radius RD (kT/?)½
E Ue
screened
E
RD
0
Rn
Rt
Ue electron screening potential
Similarly

• in terrestrial laboratories
• interaction between ions (projectiles) and atoms
  or molecules (target)

Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
42
Experimental approach
electron screening
cross-section enhancement factor
BUT electron screening in lab DIFFERENT from
electron screening in plasma
need to understand flab(E)
improve calculation of fplasma(E)
PROBLEM experimental Ue gtgt theoretical Ue
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
43
Experimental approach
general features
Explosive burning stages of stellar evolution
T gt 108 K ? E0 1 MeV Ecoul ?
10-6 barn lt ? lt 10-3 barn ? NO extrapolation
needed ? average interaction time ?
lt?vgt-1 seconds unstable species DO
GOVERN nuclear processes
FEATURES
? lt?vgt-1 seconds ? unknown nuclear
properties ? low beam intensities
(5-10 o.d.m. lower than for stable beams) ?
beam-induced background
PROBLEMS
unstable species ? RIBs production and
acceleration ? large area detectors ?
high detection efficiency
REQUIREMENTS
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
44
Experimental approach
data needs
NUCLEAR DATA NEEDS
knowledge required
reactions involving A lt 30 A gt 30
cross-section dependence individual
resonances nuclear properties statistical
properties Hauser-Feshbach calculations
excitation energies spin-parity widths decay
modes
masses level densities part. separation energy
experimental constraints wherever possible
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
45
Experimental approach
explosive nucleosynthesis
EXAMPLES
rp-process
r-process
rapid proton captures X(p,?)Y
rapid neutron captures X(n,?)Y
synthesis of neutron-rich nuclei A gt 60
synthesis of proton-rich nuclei A 100
proton capture
neutron capture
?- decay
? decay
Z
stable
N
unstable
(Shotters lecture)
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
46
Overview of main astrophysical processes
M.S. Smith and K.E. Rehm, Ann. Rev. Nucl. Part.
Sci, 51 (2001) 91-130
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
47
Thermonuclear reactions in stars
overview
Hydrostatic equilibrium T 106 - 108 K ?
average interaction time ? lt?vgt-1 109 y
unstable species do
not play significant role
stellar reactions take place through TUNNEL effect
kT ltlt E0 ltlt Ecoul
10-18 barn lt ? lt 10-9 barn
Extrapolation NEEDED Solutions underground
measurements indirect approaches BUT!
Electron screening problem
Explosive phenomena T gt 108 K ?
average interaction time ? lt?vgt-1 seconds
unstable nuclei govern
nuclear reaction processes

• Sophisticated techniques for
• RIBs production and acceleration
• Ad-hoc detection systems required

E0 Ecoul
10-6 barn lt ? lt 10-3 barn
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003
48
Further reading
W.D. Arnett and J.W. Truran
Nucleosynthesis The University of
Chicago Press, 1968 J. Audouze and S. Vauclair
An introduction to Nuclear Astrophysics
D. Reidel Publishing Company, Dordrecth,
1980 E. Böhm-Vitense Introduction to
Stellar Astrophysics, vol. 3 Cambridge
University Press, 1992 D.D. Clayton
Principles of stellar evolution and
nucleosynthesis The University of
Chicago Press, 1983 H. Reeves Stellar
evolution and Nucleosynthesis Gordon
and Breach Sci. Publ., New York, 1968 C.E. Rolfs
and W.S. Rodney Cauldrons in the
Cosmos The University of Chicago Press,
1988 (the Bible)
Copies of this lecture at www.ph.ed.ac.uk/malio
tta/teaching
Second European Summer School on Experimental
Nuclear Astrophysics - St. Tecla, Sept. 28th
Oct. 5th 2003