1. Azarquiel School-june,11-2017
Part II: Nucleosynthesis and evolution
1. Basics of Thermonuclear Reactions
2. Burning Processes
3. s-process
4. r-process
Azarquiel School-june,

2. 1. Basics of Thermonuclear Reactions
1.1 Cross section and reaction rate
Definition of cross section
Let: N x=number density of type x
N a =number density of type a
Reaction between a and X is notified as:
Reaction rate:
Effective area
Particle flux
But in stars we deal with plasma in which the particles have a spectrum of Velocities:
We need the average of :

3. 2 particles are destroyed when a=X) Equating both expression
The reaction rate is:
To avoid double counting in case a=X, we write:
Lifetime  a (X):
This is a rate of change of NX when reacting with a
Definition:
But
(to take into account that
2 particles are destroyed when a=X)
Equating both expression
to obtain:
If a nucleus X is destroyed by several reaction:

4. What is the form of the velocity spectrum
Nuclei in the star’s plasma are certainly non-degenerate owing to their high masses. In thermodynamic equilibrium their velocity distribution is described by the Maxwell-Boltzmann distribution:
Note that
Is the reduced mass, since we are considering the center of mass system

5. The Gamow Peak
And inserting (v) as above, we get:
Using we get

6. The Gamow Peak
Maxwell-Boltzmann
Distribution
GAMOW PEAK (magnified)
Tunnel Effect
through
Coulomb
Barrier
E0
KT=1-3 KeV
E0 =15-30 KeV
Effective burning energy
This is the secret of thermonuclear reactions in stars without it no human being can exist.
Most of the reactions occur in the high-energy tail of the Maxwell-Boltzmann
distribution . The fact that E0 >> KT is due to the penetration factor pushing it
to higher energy.

7. In case of the non-resonant reactions , the S(E) factor is slowly varying and can be taken approximately as S(E)=S(E0) and pulled outside the integral. In other words, the product of the exponentals lead to the Gamow peak obviously.
Taking the first derivatives of the integrand, one can find the maximum of the Gamow peak at:
Examples:
T6=15
Reaction
E0 (keV)
p+p
P+14N
+12C
16 O+ 16 O
Effect of the coulomb barrier is clearly seen !

8. 2. Burning Phases Chain I Borexino Expr. Qeff=26.20 Mev Chain III
2.1 Hydrogen Burning, proton-proton chain
The power of the Sun and solar like stars up to 1.5 MSun
Gallax neutrino experiment
71Ga
T=(10-15)x106 K
86%
14%
14%
0.02%
Chain I
Borexino
Expr.
Qeff=26.20 Mev
Chain III
Net effect:
Chain II
Qeff=25.66 Mev
Davis neutrino experiment
Qeff=19.17 Mev

9. Ne-NA cycle and Mg-AL cy=ycle Dr Sneden will talk about them.
The CNO cycle in stars of M>1.5 MSun
Energy generation rate mainly controlled by this slowest reaction in the CN cycle:
P+gmma reactions followed by beta-plus decay
Extension of the CNO:
Ne-NA cycle and Mg-AL cy=ycle
Dr Sneden will talk about them.
near T= K
See also El Eid & Champgne , ApJ 451,298 (1995)-,

10. PP chain versus CNO cycle
M  1.5 MSun
M  1.5 MSun
CNO cycle dominates above 18x106 K

11. Helium burning: The process of carbon creation
ONLY IN STARS
Resonance
The resonance at Mev in the compound nucleus has saved our existence in this universe

12. The Helium Fusion Thru Resonance 7.654 Mev Make 16O Survival of 16O
Two-step
Process
Make 12C
Thru
Resonance
7.654 Mev
Make 16O
Survival of
16O

13. Making the Heavy Elements
abundance curve of heavy elements: solar system
Notice the shifting of the peaks of the s and r processes

14. This is also representing r-process abundances in solar system
r-process more lovely?

15. Thus there are three neutron-capture processes: s, r and s+r
Amazing & Interesting
There are only 27 pure r-nuclei and 28 pure s-nuclei. all othes s+r nuclei
Thus there are three neutron-capture processes: s, r and s+r
Illustration:
See next slide to see why s, r an S+r

16. s process
(n,) reactions increases the mass number (A) . However, if the isotope is unstable, then subsequent process depends on the neutron flux and the life time ()
Standard s-process : beta decay wins the game
The produced nuclei are closed to the valley of beta stability
radioactive
s sr s sr sr r r
See next slide

17. r-process: neutron capture wins the gam
This process operates far from the valley of stability
Beta decay lifetimes: ( ) s.
For n=10-4 s : Nn=3x1020 neutron/cm3

18. r-process 2.2 s and r processes
In the Galaxy, heavy elements beyond copper are formed
by 2 neutron-capture processes:
Primary process: built up from scratch.
s-process
r-process
1/2 & 1/2
“s” for slow, but what is slow?:
Slow is the n-capture
compared to beta-decay
“r” for rapid, but what is rapid?:
neutron capture much faster compared to beta-decay
Standard scenario::
Core collapse SNe
M >8 Msun.
Detailed model not yet done
Main component
A > 90,
AGB stars:
(1-8) MSun
Weak component
A < 90, Massive stars,
M15 M Sun
Both processes very challenging on theoretical an experimental ground
Secondary process:
Depends on iron seed-abundance

19. 2.3 Details of the s-process
Whenever neutrons are produced in stellar environment, they become quickly ( s) thermalized via elastic scattering in the star’s plasma. They obey the Maxwell-Boltzmann distribution (MBD):
From lab experiments, we know that the neutron-capture cross section varies like:
S-wave scattering
MBD
E0 =most probable energy
E0=kT
Neutron energy En

20. (n,) - decay Neutron sources T-range Evolutionary phase
KeV or 1.5 – 3.5 x 108
He-burning: AGB stars &
90 KeV or 109 K
Helium burning and Shell -Carbon burning in massive stars. Why not core carbon burning? Ask me if u don’t know
(n,) Z
Z+1 -
Radioactive
A -1
(n,)
- decay Z A
A+1
Time variation of abundances
of stable isobar NA: N
(Z,A-1)
(Z.A)
(Z.A+1)
creation
destruction

21. = neutron flux [neutron/(cm2 .s)]
Along the s-process path, a (n,gamma) reaction increases the mass number A, but if the isotope is unstable, then the subsequent process depends on the neutron flux (or neutron density) and the beta-minus half-life time compared to the neutron capture lifetime.
Beta-rate in stellar plasma Time-dependent
Two limiting cases:
Radioactive nuclei decay very fast, so that their abundances can be neglected
(a)
(b)
Radioactive nuclei can be treated as stable nuclei
Temperature does not change fast during s-process:
Measuring  near VT is a good approximation.
= neutron flux [neutron/(cm2 .s)]

22. define neutron exposure:
using
Self-regulating equation with a steady-state solution:
Main component
weak
Massive stars
AGB stars
Weak component
Two components
This indicates:
a nucleus with small  would have large abundance, and a nucleus with high  would have small abundance.
 [mb] N/(106 Si)
As one sees, this curve provides the link to the stellar evolution
Reference: Käppeler etal, APJ, 257, 821, (1982)

23. Implication of the N-curve
 Obtaining the abundances of the r-nuclei (see below)
The fact that N is not constant all the way led to the insight that two component exist that lead to the s-process, and to the insight that the thermal pulsations in AGB stars are required for the production main component. We come back to this later

24. Testing branching along the s-process path
93Zr with long lifetime (1.5 My), while 95Zr has 64 day. For neutron density < 3x108 cm-3 , 95Zr decays into 95Mo over 95Nb. For higher neutron density, 95Zr(n,)96Zr becomes effective and leads to subsolar ratio of 96Zr/ 94Zr found in Sic grains. It seems (See Lugaro et al, 2003) that is needed as a neutron source
Mo= Molybdenum
Nb=Niobium

25. s-process in massive stars
 In core He-burning, s-process rather sensitive to the reaction
s-process in Carbon-shell burning depends on the evolution of massive stars during core He-burning and the ensuing carbon burning. In particular, on the location of the carbon shells. This is related to the mass fraction of carbon (X12) determine by the
operating in late phase of core He-burning
If X12 is relatively high (Kunz et al rate), then the s-process occurs earlier in the evolution (before Ne burning. Consequently, lower neutron density is achieved, and only the Sr-region is reached- that is amazing
If X12 is relatively low, the s-process occurs later during Ne-burning . This leads to higher neutron density in the range of 1011 n/cm-3. Consequently, some production of 90Zr is possible.

26. 3.2 Massive stars Why the difference? See next
Darker areas: CONVECTIVE
see
El Eid, Meyer, The
APJ 611, 452 (2004)
The, El Eid, Meyer:
APJ, 655, 1058, (2007)
Note that the energy production does not comprise the whole convective core
NACRE
NACRE
Why the difference?
See next

27. Nuclear physics essential for understanding the structure of stars
It is the rate
Nuclear physics essential for understanding the structure of stars
He burning;15-30 MSun
CF85: Caughlan et al ; Atom.. Data Nucl. Data Tables, 32, 197 (1985)
Kunz: Kunz etal. APJ, 567, 643 (2002)
NACRE: Angulo et al , Nucl. Phys. A, 656, 3A, 1999
Buchmann: 1996, APJ, 468, L127 (1996)
june 11, 2017

28. Limongi, Straniero , Chieffi APJS, 129, 625 (2000)
25 Msun star
Imbriani et al (2001), ApJ 558, 903
Rate of
CF85 > CF88
No convective
carbon-burning
core
CF85
X 12=0.18
X12 lower
Remaining
car bon mass fraction
No convective core
But here
CF88
X12=0.42
See also :
Limongi, Straniero , Chieffi
APJS, 129, 625 (2000)
june 11, 2017

29. Learn Effect
The carbon burning is an important evolutionary phase in massive stars:
It is a branching point between the formation of white dwarfs and the
evolution toward core collapse supernovae
It influences strongly the subsequent evolution, especially through carbon-shell
burning, where the s-process nucleosynthesis operates in part
It depends on several factors:
Mass of star
Carbon mass fraction left over from Helium burning, thus on the rate of
Neutrino losses
june 11, 2017

30. 25 M_sun
Details: see The, El Eid, Meyer: APJ, 655, 1058 (2007)

31. 25 M_un

32. s-process outcome Solar abundances
Main component
Weak
component
Massive
Stars
AGB Stars
Overproduction factors of the IMF-averaged stars (15,20,25,30) Msun

33. Get in touch with Galactic Evolution
Disk stars
Filled circles:
Halo stars
Log(57La/ 63Eu)
Hallo stars
Lanthanum/ Europium
Filled diamonds:
Disk stars
S-process versus
r-process
prediction:
r-process
only
[Fe/H]
Metallicity
1. La/Eu increases as the s-process starts contributing, at about [Fe/H]  -2
metal-rich disk stars have larger ratio than halo stars
2. Only metal-poor stars have ratios consistent with the r-process
3. The s-process seems to have contributed at earlier phase ([Fe/H]<-2) , but how? In which stellar mass range and at what metallicity?
[A/B] = log(NA/NB)* - log(NA/NB)

34. 2.3 Some details on the r-process
Abundances: Nr  N -f(A)/(A)
Taking advantage that  Ns falls on the flat part of the curve
Three abundance peaks
Result:

35. The above characteristics and the fact that the ntural radioactive nuclei
232Th- 235U- 238U
cannot be produced by the s-process led the concept of the r-process (B2FH57)
Home work
For the students
Why the s-process cannot produce the super heavy radio active nuclei?
Basically:
The r-process is based on a flow concept similar to the s-process, but:
High neutron density is needed (>1020 neutrons/cm 3, such that n<< 
(neutron capture wins against beta-decay most od the time, but not always)
Effect:
A nucleus (Z.A) becomes more neutron rich due to (n,) reactions.
neutron number increases, addition of neutrons stops when binding energy Qn of the neutrons approaches zero
It can be shown (see Rolfs & Rodney, “Cauldrons in the Cosmos”, 1988) that this occurs for given temperature T and neutron density Nn,
when the (n, ) rate and (, n ) rate are in equilibrium

36. For Nn=1024 cm-3 and T=109 K , both rates equal when Qn =2 MeV
For Nn=1024 cm-3 and T=109 K , both rates equal when Qn =2 MeV. The chain of (n,) stops and beta-decay becomes effective.
Then:
If neutron capture stops, what will the nuclei do?
Well, they wait for beta-decay. This is seen by the step-like structure in the r-process path

37. Consequence: Compare:
This is called “waiting point” for each charge Z. This means beta-decay must occur before the next neutron capture can proceed, and this defines the r-process path shown above.
It is then clear that abundances are not characterized by NA as in the s-process, but by NZ
The time dependence of the abundances is give by a set of coupled differential equations:
Where Z is the beta-decay rate at the waiting point. This set of equations have the tendency to reach equilibrium (dNZ/dt=0) with NZ  1/Z=
Consequence:
Abundances in the r-process correlate with beta-decay lifetimes at waiting points
Compare:
In the s-process the abundances are correlated with the capture cross section:
NA1/A1/A

38. Next
The r-process transforms the matter to the neutron side. However, if the path reaches the magic numbers: Nm =50,82,126
Then, Qn becomes very small and  long. One has special waiting points
The next heavier isotope with Nm+1 neutrons undergoes beta-decay to (Z+1,Nm)
It is again a nucleus with the magic number (Nm) .
The nuclei approaches the valley of stability, so that the normal sequence proceeds
After termination the n-rich nuclei undergo beta-decay keeping the mass number the same
N=82

39. The insight - s.r Progenitor N-magic Product Non-magic
278 ms
s.r
Product
Non-magic
Relatively high abundance
See Fig. on p. 9
Progenitor
N-magic

40. The longer than average beta-decay lifetimes of the of these special nuclei explain their large abundances.
Since their associated proton numbers are smaller than in the s-process, isobars of the r-process are shifted as we have seen earlier
Comparison between s -process and r- process
s-process r-process
Neutron density
neutrons/ cm3)
Abundances n-capture cross beta-decay time
correlated with cross section at waiting points
N=const no such relation
Astrophysical Site AGB stars Supernovae
massive stars Neutron star mergers
Basic character Secondary process Primary process comes later early happening
in galaxy in supernovae