Lecture 20: Astrophysical Reaction Ratesinpp.ohio.edu/~meisel/PHYS7501/file/Lecture20...is the...
Transcript of Lecture 20: Astrophysical Reaction Ratesinpp.ohio.edu/~meisel/PHYS7501/file/Lecture20...is the...
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Lecture 20: Ohio University PHYS7501, Fall 2017, Z. Meisel ([email protected])
Lecture 20: Astrophysical Reaction Ratesâ¢Evidence for nuclear reactions in spaceâ¢Thermonuclear reaction ratesâ¢Non-resonant ratesâ¢Resonant ratesâ¢Reaction networks
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Birth of Nuclear Astrophysicsâ¢Nuclear astrophysics was spawned in 1920 by a realization of Arthur Eddington (The Observatory (1920):â¢Based on fossil evidence at the time, the Earth was known to be more than several hundred million years old and the sun presumably had to be at least as old
â¢A plausible explanation for the sunâs power might be gravitational energy being converted into heat from the gaseous solar sphere collapsing, taking place on the Kelvin-Helmholtz timescale
⢠ðððŸðŸðŸðŸ = ðððððððððð ðŸðŸðŸðŸðŸðŸðŸðŸðððŸðŸðŸðŸ ðžðžðŸðŸðŸðŸðžðžðžðžðžðžðžðžðŸðŸðŸðŸðžðžðžðžðžðž ð¿ð¿ððð¿ð¿ð¿ð¿ ð ð ðððððŸðŸ
= (ðððððððŸðŸðŸðŸðððŸðŸðððð ðžðžðŸðŸðŸðŸðžðžðžðžðžðž)/2ð¿ð¿ð¿ð¿ð¿ð¿ðŸðŸðŸðŸððð¿ð¿ðŸðŸðððžðž
= ðºðºðð2
2ð ð ð¿ð¿â 16ðððððð for the sun
â¢Instead, maybe itâs just a lump of burning coal!⢠Typical chemical bond energies are ~eV⢠The sun has a mass of ~1030kg and a nucleus is ~10-27kg, it has ~1057 nuclei (or atoms)⢠Since the sun releases ~1039MeV/s, chemical burning would last roughlyð¡ð¡ððð¿ð¿ðžðžðŸðŸ = ðŽðŽð¿ð¿ððð¿ð¿ðŸðŸðð ðððð ð¹ð¹ð¿ð¿ðŸðŸðð
ð ð ðððððŸðŸ ðððð ð¹ð¹ð¿ð¿ðŸðŸðð ðµðµð¿ð¿ðžðžðŸðŸðŸðŸðŸðŸðžðž~ 1057ððððððð¿ð¿ð¿ð¿â1ðŸðŸðð/ððððððð¿ð¿1039ðððŸðŸðð/ð¿ð¿â106ðŸðŸðð/ðððŸðŸðð
~1012ð ð ~30ðððððð⢠Third timeâs the charmâŠletâs look at nuclear energy
⢠Aston measured a 32MeV discrepancy between 4 protons and 1 Helium nucleus (4p->α actually yields ~27MeV)⢠Multiplying our fuel amount by 106 (eV->MeV) results in ~10ðºðºðððð burn time âŠwhich finally does the job
2
from the virial theorem
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Evidence for recent nuclear reactions in space (selected examples)
3
Solar Μ attributable to hydrogen burning sequences
Bahcall, Serenelli, & Basu, ApJL (2005)
γ-rays of short-lived isotopes (e.g. 44Ti) in supernova remnants
B. Grefenstette et al. Nature (2014)
Anomalous isotopic ratios in meteoritic âpre-solarâ dust grains
N.Liu et al ApJL (2017)
Neutron star merger afterglow associated with radioactive decay
Berger, Fong, & Chornock ApJL (2013)
Radioactive elements (e.g. Tc) in stellar spectra
B. Peery PASP (1971)
Match in energetics(e.g. C-fusion in X-ray superbursts)
E.F. Brown ApJL (2004)
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Reaction rateâ¢The way nuclei influence and are influenced by astrophysical environments is through energy generation and element transmutation (ânucleosynthesisâ). The importance of a pair of nuclides is going to be related to the rate at which they react.
â¢We can figure out what that reaction rate is by considering a simple example to a gas with two species, each with their own number density, inside a volume at some temperature
â¢For simplicity, weâll consider one species as the projectile and the other as the target(though it doesnât matter since all of our calculations will be in the center of mass)
â¢The reaction rate between species 1 and 2 will be:ð ð ð ð ð¡ð¡ð ð = ð¹ð¹ð¹ð¹ð¹ð¹ð¹ð¹ ðððð 1 â ððð¹ð¹ððððð ð ðð ð·ð·ð ð ð·ð·ð ð ð·ð·ð¡ð¡ðð ðððð 2 â (ðŒðŒð·ð·ð¡ð¡ð ð ððð ð ðŒðŒð¡ð¡ð·ð·ððð·ð· ððððððððð ð ððð·ð·ð¹ð¹ð·ð·ð¡ð¡ðð ðððððð 1 + 2)
â¢1âs flux is the number of 1 atoms per area per time: ð¹ð¹1 = ð·ð·1ð£ð£where ð·ð·1 is the 1 number density and ð£ð£ is the relative velocity
â¢The interaction probability is provided by the cross section ðð12(ð£ð£)â¢So, ðð ð£ð£ = ð¹ð¹1ð·ð·2ðð12 ð£ð£ = ð·ð·1ð·ð·2ðð12 ð£ð£ ð£ð£, is the reaction rate for velocity ð£ð£
4
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Thermonuclear reaction rate⢠Of course, nature isnât so neat and orderly. Any environment will have a finite temperature and
therefore any nuclei will have a velocity distribution ðð(ð£ð£) thatâs defined by the temperature.⢠To get an average rate given some relative velocity distribution, we need to do an average
ðð = ï¿œ0
âðð(ð£ð£)ð·ð·1ð·ð·2ðð12 ð£ð£ ð£ð£ ððð£ð£ = ð·ð·1ð·ð·2 ï¿œ
0
âðð(ð£ð£)ðð12 ð£ð£ ð£ð£ ððð£ð£ = ð·ð·1ð·ð·2 ððð£ð£ 12
(If the reaction is endothermic then the integral lower-limit is the reaction threshold)
⢠To avoid double-counting of particles, we have to make a slight modification for identical nuclei:ðð = ðŸðŸ1ðŸðŸ2 ðððð 12
1+ð¿ð¿12, where ð¿ð¿12 is the Kronecker delta
⢠Determining the reaction rate between a pair of particles ððð£ð£ is the nuclear physicistâs task⢠The number densities are determined be calculating the effects of all reaction rates as a
function of time in a reaction network to see how many nuclei are created/destroyed⢠Itâs more common to deal with matter densities and get the number density by taking into
account the fraction of mass species ð·ð· contributes: ð·ð·ðŸðŸ = ðððððŽðŽðððð
ð¿ð¿ðððððððð,ðð= ðððððŽðŽðððŸðŸ
where ðððŽðŽ is Avogadroâs number, ððð¿ð¿ðððð,ðŸðŸ is the molar mass (a.k.a. mass in amu),and ðððŸðŸ (†1) and ðððŸðŸ are the mass and mole fractions, respectively 5
Similarly, the reduced reaction rate NA<Ïv> is often used
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Thermonuclear rates and the Maxwell-Boltzmann distribution⢠Typically, the environment weâre considering is a classical gas in thermodynamic equilibrium,
so the velocity distribution is described by the Maxwell-Boltzmann distribution
ðððððµðµ ð£ð£ = 4ððð£ð£2 ð¿ð¿2ðððððµðµðð
3/2exp â ð¿ð¿ðð2
2ðððµðµðð, where ðððµðµ is the Boltzmann constant and the pre-
factor is for normalization (i.e. â«0âðððððµðµ ð£ð£ ððð£ð£ = 1)
⢠Both species interacting in an environment will have this distribution, so taking velocities for1 and 2 in the lab-frame, ððð£ð£ 12 = â«0
ââ«0âðððððµðµ ð£ð£1 ðððððµðµ ð£ð£2 ðð12 ð£ð£ ð£ð£ððð£ð£1ððð£ð£2
⢠This is less-than desirable, since we would rather be working in terms of relative velocity only,so we consider the fact that ð£ð£ðŸðŸ = ð£ð£ðŸðŸð¿ð¿ ± ð£ð£
⢠This means the double integral can be re-stated in terms of integrating over ð£ð£ and ð£ð£ðŸðŸð¿ð¿:ððð£ð£ 12 = â«0
ââ«0âðððððµðµ ð£ð£ðŸðŸð¿ð¿ ðððððµðµ ð£ð£ ðð12 ð£ð£ ð£ð£ððð£ð£ðŸðŸð¿ð¿ððð£ð£
⢠Since the cross section only depends on the relative velocity and â«0âðððððµðµ ð£ð£ðŸðŸð¿ð¿ ððð£ð£ðŸðŸð¿ð¿ = 1,
ððð£ð£ 12 = ï¿œ0
âðððððµðµ ð£ð£ ðð12 ð£ð£ ð£ð£ ððð£ð£
where we should keep in mind ðð in ðððððµðµ refers to the reduced mass ðð6
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Thermonuclear rates and the Maxwell-Boltzmann distribution⢠Inserting ðððððµðµ ð£ð£ into ððð£ð£ 12 = â«0
âðððððµðµ ð£ð£ ðð12 ð£ð£ ð£ð£ ððð£ð£ yields
ððð£ð£ 12 = 4ðð ðð2ðððððµðµðð
3/2â«0âðð12 ð£ð£ ð£ð£ ð£ð£2exp â ðððð2
2ðððµðµððððð£ð£
⢠Noting the center of mass energy ðžðž = 12ððð£ð£2 and so ðððžðž = ððð£ð£ ððð£ð£ (i.e. ð£ð£2 â 2
ðððžðž, ððð£ð£ â 1
ðððððððžðž),
we finally arrive at a useful equation for the astrophysical reaction rate
ððð£ð£ 12 =8ðððð
1ðððµðµðð 3/2 ï¿œ
0
âðð12 ðžðž ðžðžexp â
ðžðžðððµðµðð
ðððžðž
⢠Note that this is the general formula for classical gases.Weâll go over special cases (for classical gases) in a bit.
⢠As an aside, personally I find ðððµðµ hard to remember,but I find it easier to remember that 11.6045 â ðžðžðððŸðŸðð = ðð9,where ðžðžðððŸðŸðð is energy in MeV and ðð9 is temperature in GK
7
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Reverse (a.k.a. inverse) rates⢠Even when a particular reaction direction is exothermic (ðð > 0), for a finite temperature,
some fraction of particles will have energies with ðžðž ⥠ðð, and that fraction will grow with ðð⢠Recall that the cross section for an entrance channel through a compound nuclear state is the
product of the effective geometric area of the projectile ( λ122ðð
2), the number of sub-states that
can be populated (2ðœðœ + 1), the probability of being in a particular sub-state for each of the reactants ( 1
(2ðœðœ1+1)1
(2ðœðœ2+1)), a factor of 2 for identical particles (1 + ð¿ð¿12), and the matrix elements
for forming the compound nucleus ð¶ð¶ through 1 + 2 and decaying via 3 + 4:
ðð12â34 = ððλ122ðð
2 2ðœðœ + 12ðœðœ1 + 1 2ðœðœ2 + 1
(1 + ð¿ð¿12) 3 + 4 ð»ð»2 ð¶ð¶ ð¶ð¶ ð»ð»1 1 + 2 2
⢠The reverse process would then be
ðð34â12 = ððλ342ðð
2 2ðœðœ + 12ðœðœ3 + 1 2ðœðœ4 + 1
(1 + ð¿ð¿34) 1 + 2 ð»ð»1 ð¶ð¶ ð¶ð¶ ð»ð»2 3 + 4 2
⢠I.e. theyâre identical except for the statistical and geometric factors out front
8
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Reverse (a.k.a. inverse) rates⢠Making the substitution that
λðððð2ðð
2= â2
ðð2= â2
2ðððððððžðžððððand noting ðððŸðŸðð = ð¿ð¿ððð¿ð¿ðð
ð¿ð¿ðð+ð¿ð¿ðð,
taking the ratio of the forward and reverse cross sections yieldsðð12ðð34
=ðŽðŽ3ðŽðŽ4ðŽðŽ1ðŽðŽ2
ðžðž34ðžðž12
2ðœðœ3 + 1 2ðœðœ4 + 12ðœðœ1 + 1 2ðœðœ2 + 1
(1 + ð¿ð¿12)(1 + ð¿ð¿34)
where ðŽðŽðŸðŸ can be used instead of ðððŸðŸ since the units cancel
⢠For the rates, recall ððð£ð£ 12 = 8ðððð12
1ðððµðµðð 3/2 â«0
âðð12 ðžðž12 ðžðž12exp â ðžðž12ðððµðµðð
,
so ððð£ð£ 34 = 8ðððð34
1ðððµðµðð 3/2 â«0
âðð34 ðžðž34 ðžðž34exp â ðžðž34ðððµðµðð
⢠Since ðžðž34 = ðžðž12 + ðð12 (if ðð12 > 0), when we take the ratio ððð£ð£ 34/ ððð£ð£ 12 , the integrands mostly cancel except for the energy-independent part:
ððð£ð£ 34
ððð£ð£ 12=
2ðœðœ1 + 1 2ðœðœ2 + 12ðœðœ3 + 1 2ðœðœ4 + 1
(1 + ð¿ð¿34)(1 + ð¿ð¿12)
ðð12ðð34
3/2
exp âðð12ðððµðµðð
⢠The exponential dominates, so ðððð 34ðððð 12
â exp â ðð12ðððµðµðð
gives the correct order of magnitude9
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Reverse (a.k.a. inverse) rates: photodisintegration⢠For photodisintegration, we have to take into account the fact that the photons follow a
Planck distribution and not a Maxwell-Boltzmann distribution
⢠So the reverse rate becomes: ðððð 3γ
ðððð 12â ðð12ðŸðŸ2
ðððµðµðð
3/2exp â ðð12
ðððµðµðð
⢠This extra factor is actually a huge deal because of that extra temperature dependence⢠In fact, for low ðð-value reactions,
the photodisintegration rate is dominant
10
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Aside: S-factors, the nuclear physics nuggets of ððð£ð£
⢠Often itâs useful to remove the trivial energy dependence from the cross section, in particular for charged-particle reaction rates
⢠The idea is that ðð(ðžðž) contains all of the interesting physics
⢠Since the energy dependence is different for different types of reactions,e.g. direct capture of a neutron as compared to direct capture of a charged particle, the factorization that is done to get ðð(ðžðž) depends on the reaction type
11
Rolfs & Rodney, Cauldrons in the Cosmos (1988)
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Aside-er: Energy release in a reaction⢠How much energy does a nuclear reaction rate release into the environment?⢠This is as simple as it sounds, where the rate of energy release is just the product of the energy
release per reaction times the reaction rate: ðð12 = ðð12ðð12⢠Given our pre-agreed upon units, ðð = MeV
cm3s,
âŠthough some people prefer energy released per gram: Ìðð12 = ðð12ðžðž12ðð
(units MeV/(g.s))
⢠To be on the safe side, one needs to keep in mind the reverse rate,since we saw it can be significantat high temperatures: ðð12,ðŸðŸðŸðŸðð = ðð12 + ðð34 = ðð12(ðð12âðð34)
12
Wiescher & Schatz, Prog.Theory.Phys.Suppl. (2000)
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Thermonuclear rate: Direct neutron-capture⢠Recall that at low energies, those of interest for nuclear astrophysics, the neutron-capture
cross section is described by the 1/ð£ð£ law: ðððŸðŸ ðŸðŸðððð â1ðððð
⢠As such, itâs clear that ððð£ð£ ðŸðŸ ðŸðŸðððð â ðŒðŒððð·ð·ð ð ð¡ð¡ð ð ð·ð·ð¡ð¡â¢ The cross section (and rate) can be characterized by the S-factor at thermal energies and any
deviation from 1/ð£ð£ behavior is accounted for by the local derivative(s) of the S-factor:
ðð ðžðž =ðð2ðžðž
ðð 0 + ï¿œÌï¿œð 0 ðžðžÂœ + 12ï¿œÌï¿œð 0 ðžðž + â¯
⢠ðð(0) is generally the S-factor at thermal energy (ð£ð£ððð¡ = 2.2 à 105 ðŸðŸð¿ð¿ð¿ð¿
= ðžðžð¿ð¿ = 2.53 à 10â8ððð ð ðð):ðð 0 = ððððð¡ð£ð£ððð¡ = 2.2 à 10â19ððððð¡ðððð
3
ð ð , where ððððð¡ is the thermal cross section in barns
⢠ᅵÌï¿œð 0 and ï¿œÌï¿œð 0 are fit to cross section data near thermal energies
⢠Employing this ðð ðžðž in the general equation ððð£ð£ = 8ðððð
1ðððµðµðð 3/2 â«0
âðð ðžðž ðžðžexp â ðžðžðððµðµðð
ðððžðž,results in
ððð£ð£ ðŸðŸ ðŸðŸðððð = ðð 0 + 4ððï¿œÌï¿œð 0 ðððµðµðð 1/2 + 3
4ï¿œÌï¿œð 0 ðððµðµðð + â¯13
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Thermonuclear rate: Direct neutron-capture
14
Examples (from the ):
3He(n,p) 32Si(n,γ) 208Pb(n,γ)
Clear that the main feature is ððð£ð£ ðŸðŸ ðŸðŸðððð â ðŒðŒððð·ð·ð ð ð¡ð¡ð ð ð·ð·ð¡ð¡ over 3 orders of magnitude in T
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Thermonuclear rate: Direct charged particle-capture⢠Recall that the cross section for charged-particle capture depends on the effective geometric
area of the projectile, described by λðððŸðŸðµðµðžðžðððžðžðððŸðŸðŸðŸ, and the probability of the charged projectile tunneling through the Coulomb barrier of the target
⢠When discussing α decay, we showed ððððð¿ð¿ðŸðŸðŸðŸðŸðŸðð ðžðž = exp(â2ðððð),
where the factor with the Sommerfeld parameter is 2ðððð = 2ðð ðŸðŸ2
âðŸðŸðð1ðð2
ðððŸðŸ2
2ðžðž
⢠For ðžðž in MeV and ðŽðŽ in atomic mass units (1u = 931.5MeV/c2):
2ðððð = 0.989ðð1ðð21ðžðž
ðŽðŽ1ðŽðŽ2(ðŽðŽ1+ðŽðŽ2)
⢠Removing the trivial energy dependency:ðððŸðŸð¡.ðŸðŸðððð(ðžðž) = 1
ðžðžexp â2ðððð ðð(ðžðž)
15Rolfs & Rodney, Cauldrons in the Cosmos (1988)
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Thermonuclear rate: Direct charged particle-capture⢠Employing Ï(ðžðž) = 1
ðžðžexp â2ðððð ðð(ðžðž) in ððð£ð£ = 8
ðððð1
ðððµðµðð 3/2 â«0âðð ðžðž ðžðžexp â ðžðž
ðððµðµðððððžðž gives:
= 8ðððð
1ðððµðµðð 3/2 â«0
â ðð ðžðž exp â ðžðžðððµðµðð
â 2ðððð ðððžðž
⢠The integrand is a product of the probability of a charged-particle pair having energy ðžðž(from the Maxwell-Boltzmann distribution) and the tunneling probability for that energy
⢠The integrand maximum (found by solving for the derivative being equal to zero) is:
ðžðžðºðº = 0.122 ðð12ðð22ðŽðŽ1ðŽðŽ2
(ðŽðŽ1+ðŽðŽ2)ðð92
1/3MeV
⢠Approximating the integrand as a Gaussianresults in a distribution with the 1/ð ð widthâðºðº= 4 ðžðžðºðºðððµðµðð
3MeV
⢠ðžðžðºðº is the Gamow peak and âðºðº is the widthof the Gamow window, which is roughlythe energy range for which we care abouta charged particle reaction rate for some ðð
16
Rolfs & Rodney, Cauldrons in the Cosmos (1988)
Note: Donât be too naïve when using the Gamow window estimate.Itâs based on a roughly constant S(E), so the true window of interest could be different (T.Rauscher PRC 2010).
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Thermonuclear rate: Direct charged particle-capture⢠The S factor is again represented with a Taylor expansion ðð ðžðž = ðð 0 + ï¿œÌï¿œð 0 ðžðž + 1
2ï¿œÌï¿œð ðžðž ðžðž2 + â¯
⢠As with neutron-capture, it is determined experimentally. Here, by dividing the trivial energy dependence from the measured cross section as a function of energy[Or doing something equivalent with a fancy R-matrix fit, e.g. A.Kontos et al PRC2013)]
⢠The energy dependent terms of ðð(ðžðž) winds up making the reaction rate integral have several temperature-dependent factors, which make for a rather complicated form:(See C. Iliadis, Nuclear Physics of Stars or Fowler, Caughlan, & Zimmerman, Annu.Rev.Astron.&Astro. (1967))
⢠This is the inspiration for the functional formof parameterized reaction rates used in databases,e.g. JINA REACLIB (R. Cyburt et al. ApJS (2010))
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Thermonuclear rate: Narrow resonance(s)â¢Recall the Breit-Wigner form we found for the resonant reaction cross sectionðððµðµðµðµ,ðð ðð,ðð ðð(ðžðž) = ðð λ
ðð
2 2ðœðœ+1(2ðœðœðð+1)(2ðœðœðð+1)
Îðððð(ðžðž)Îðððð(ðžðž)(ðžðžâðžðžð ð )2+(Î(ðžðž))2/4
â¢Employing this in the general form ððð£ð£ = 8ðððð
1ðððµðµðð 3/2 â«0
âðð ðžðž ðžðžexp â ðžðžðððµðµðð
ðððžðž,we realize that the contributions of the integrand are pretty negligible outside of ðžðžð ð
â¢So, we make the approximation ððð£ð£ ðžðžðŸðŸð¿ð¿ = 8ðððð
ðžðžð ð ðððµðµðð 3/2 exp â ðžðžð ð
ðððµðµððâ«0âðððµðµðµðµ ðžðž ðððžðž
â¢Noting that λ changes little over the resonance λ â λð ð , writing the statistical factor as ðð, and noting the widths Î are essentially constant over the resonance we findâ«0âðððµðµðµðµ ðžðž ðððžðž â ðð λð ð
ðð
2ÎððððÎðððð â«0
â 1ðžðžâðžðžð ð 2+(Î)2/4
ðððžðž = 2λð ð 2ððÎððððÎðððð
Î
â¢For obfuscation purposes, we substitute in ðŸðŸ for ÎððððÎððððÎ
and call ÏðŸðŸ the resonance strengthâ¢If we know the cross section at the peak of the resonance, we can make the approximation that the integral is half the width times the height: â«0
âðððµðµðµðµ ðžðž ðððžðž â ððÎðð(ðžðžð ð )
â¢The resonant rate becomes: ððð£ð£ ðžðžðŸðŸð¿ð¿ = 2ðððððððµðµðð
3/2â2 ðððŸðŸ ð ð exp â ðžðžð ð
ðððµðµðð 18
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Thermonuclear rate: Narrow resonance(s)â¢For a reaction with several resonances, the total rate is just the sum.Using units of MeV for ðžðžð ð ,ðŸðŸ and ðððŸðŸ ð ð ,ðŸðŸ and amu for ðŽðŽðŸðŸ,
ðððŽðŽ ððð£ð£ ðð ðžðžðŸðŸð¿ð¿. =1.54 à 1011
ðŽðŽ1ðŽðŽ2ðŽðŽ1 + ðŽðŽ2
ðð93/2ï¿œ
ðŸðŸ=1
ðð
ðððŸðŸ ð ð ,ðŸðŸexp â11.6045ðžðžð ð ,ðŸðŸ
ðð9cm3
mol s
â¢But how do I know which resonances matter?â¢You might guess the ones in theGamow window âŠbut it depends onthe relative widths for theexit and entrance channels.
â¢For high temperatures, particle emissionmakes the Gamow Window conceptespecially dubious
â¢Nonetheless, the GW is good first guess asto which resonances are likely important
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Iliadis, Nuclear Physics of Stars (2007)
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Thermonuclear rate: Broad resonance⢠In the past weâve specified any resonance
with Î/ðžðžð ð â³ 10% as âbroadâ⢠For such cases, we obviously need to take
into account the energy dependence ofthe widths when performing the integralover energy.See the resonant reactions lecture for adiscussion on these energy dependencies.
⢠Broad resonances are a pain becausethey make extrapolation of thereaction rate at very low energies(via the S-factor) a sketchy enterprise
⢠One approach to representthese rates analytically is as a non-resonantcontribution (for ðžðž ⪠ðžðžð ð ) plus ananalytic contribution, resulting in: ððð£ð£ ðððžðžðððððð ðžðžðŸðŸð¿ð¿ = ððð£ð£ ðððŸðŸðžðžðŸðŸðŸðŸðð + ð ð 1ððððexp â ðð2
ðððµðµðð,
where ð ð ðŸðŸ and ðð are fit to the rate calculated at several ðð by integrating over the cross section 20
Rolfs & Rodney, Cauldrons in the Cosmos (1988)
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Thermonuclear rate: Subthreshold resonance
21
Rolfs & Rodney, Cauldrons in the Cosmos (1988)
â¢Broad resonances lurking below the reaction threshold can contribute to the rate as well
â¢This can cause a huge boost over the non-resonant rateContributions from a hypothetical sub-threshold resonance with various strengths for 9Be(p,α):
E.F. Brown, ApJ (1998)
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Aside: Stellar Enhancement Factors (SEFs)â¢Itâs important to note that the rate calculated using the laboratory cross section is only for the ground-state of the target
â¢However, in a hot environment, excited states in the target will be thermally populatedâ¢The stellar enhancement factor (SEF) is the ratio of the stellar rate (which is what we want) to the rate determined from the laboratory measurement.
â¢The SEF is the ratio of theðððžðžð¹ð¹ =
ððð¡ð¡ð ð ð¹ð¹ð¹ð¹ð ð ðð ð ð ð ð ð¡ð¡ð ð ð¿ð¿ð ð ððððððð ð ð¡ð¡ðððððð ð ð ð ð ð¡ð¡ð ð =
âðððððžðžðžðžðŸðŸðð ð¿ð¿ðððððððŸðŸð¿ð¿ ð¹ð¹ððð ð ðŒðŒð¡ð¡ð·ð·ððð·ð· ðððð ððð ð ððððð ð ð¡ð¡ ððððð ð ðŒðŒð·ð·ð ð ð ð ð·ð·ð·ð· ð ð ð·ð· ðžðžð¹ð¹ðŒðŒð·ð·ð¡ð¡ð ð ðð ððð¡ð¡ð ð ð¡ð¡ð ð (ð ð ð ð ð¡ð¡ð ð ðððððð ð¡ð¡ð¡ð ð ð¡ð¡ ððð¡ð¡ð ð ð¡ð¡ð ð )ð¿ð¿ð ð ððððððð ð ð¡ð¡ðððððð ð ð ð ð ð¡ð¡ð ð
â¢The excited state populations are determined bythe partition function, and theyâre typicallyestimated using the statistical model(See e.g. A. Sallaska et al. ApJS 2013)
â¢A commonly used set of partition functions usedto calculate SEFs comes fromRauscher & Thielemann ADNDT 2000
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Charged particle reactions with Aâ€40
A. Sallaska et al. ApJS (2013)
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Total thermonuclear reaction rateâ¢Any and all of the aforementioned rate types can contribute to the overall reaction rate
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Iliadis, Nuclear Physics of Stars (2007)
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Where can I get thermonuclear reaction rates?
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REACLIB database
https://groups.nscl.msu.edu/jina/reaclib/db/
â¢The Joint Institute for Nuclear Astrophysics hosts REACLIB, which contains thermonuclear reaction rates in parameterized and tabular form
â¢Rates are based on published experimentally-constrained and otherwise purely theoretical rates
â¢REACLIB is a standard in the field,e.g. it is employed as the default in the MESA stellar modeling software
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Nuclear reaction networks
25
R.V. Wagoner ApJS (1969)
â¢Astrophysical environments typically containmany nuclei, each of which could in principleinteract with the other(in practice usually only the photons and light projectiles matter)
â¢To evaluate what happens, we need tosolve a reaction network
â¢The basic idea is to see how the abundance changesfor each isotope at each step in time based on theproduction and destruction via all mechanismsand often to include the energy generationfrom said reactions
â¢For each species ð·ð·,ðððððŸðŸððð¡ð¡
= ï¿œðð,ðð
ðððððððððððððŽðŽ ððð£ð£ ððððâðŸðŸ + ï¿œðð
λððâðŸðŸðððð â ï¿œð¿ð¿
ðððŸðŸððð¿ð¿ðððððŽðŽ ððð£ð£ ðŸðŸð¿ð¿âðððŸðŸðžðž + ï¿œðŸðŸ
λðŸðŸâðŸðŸðððð
Production Destruction
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Nuclear reaction networks
26
â¢You might consider solving the problem explicitly,stepping through time updating each ðððŸðŸ
â¢The issue is that the ordinary differential equationsðððððððððð
= ðð(ðð)weâre trying to solve are very âstiffâ,i.e. theyâre very sensitive to changes(because reaction rates have steep temperature dependencies)so we would need super tiny time-steps
â¢Instead, an implicit approach is neededâ¢We note that the change in abundance âðððŸðŸ is equalto the change in abundance at the next time-step times the time-step ðð(ðððŸðŸ(ð¡ð¡ + âð¡ð¡))âð¡ð¡
â¢The change in abundance at the next timestep is equal to the change at the present time plus the sum of the change in the change due to other speciesâ abundances changingðð ðððŸðŸ ð¡ð¡ + âð¡ð¡ = ðð ðððŸðŸ ð¡ð¡ + âðð
ðððð(ðððð ðð )ðððð
âðððð âŠwhere ðððð(ðððð ðð )ðððð
are the elements of the Jacobian matrix
â¢Solving for the change in all abundances results in âðð = ðð(ð¡ð¡)ï¿œ1âððâ ðœðœ
â1
â¢Inverting the matrix (the ~ bits) is where most of the computational cost comes in
F.X. Timmes ApJS (1999)
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Further Reading⢠Chapter 12: Modern Nuclear Chemistry (Loveland, Morrissey, & Seaborg)⢠Chapter 3: Nuclear Physics of Stars (C. Iliadis)⢠Chapters 3 & 4: Cauldrons in the Cosmos (C. Rolfs & W. Rodney)⢠Lecture Materials on Nuclear Astrophysics (H. Schatz)⢠Chapters 10 & 11: Stellar Astrophysics (E.F. Brown)⢠âThermonuclear Reaction Ratesâ, Fowler, Caughlan, & Zimmerman, Annu.Rev.Astron.&Astro. (1967)⢠R.V. Wagoner, Astrophys. J. Suppl. Ser. (1969)
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