Heavy Element Nucleosynthesispersonal.psu.edu/rbc3/A534/lec27.pdf · 2017. 6. 26. · Heavy Element...

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Heavy Element Nucleosynthesis A summary of the nucleosynthesis of light elements is as follows 4 He Hydrogen burning 3 He Incomplete PP chain (H burning) 2 H, Li, Be, B Non-thermal processes (spallation) 14 N, 13 C, 15 N, 17 O CNO processing 12 C, 16 O Helium burning 18 O, 22 Ne α captures on 14 N (He burning) 20 Ne, Na, Mg, Al, 28 Si Partly from carbon burning Mg, Al, Si, P, S Partly from oxygen burning Ar, Ca, Ti, Cr, Fe, Ni Partly from silicon burning Isotopes heavier than iron (as well as some intermediate weight iso- topes) are made through neutron captures. Recall that the prob- ability for a non-resonant reaction contained two components: an exponential reflective of the quantum tunneling needed to overcome electrostatic repulsion, and an inverse energy dependence arising from the de Broglie wavelength of the particles. For neutron cap- tures, there is no electrostatic repulsion, and, in complex nuclei, virtually all particle encounters involve resonances. As a result, neutron capture cross-sections are large, and are very nearly inde- pendent of energy. To appreciate how heavy elements can be built up, we must first consider the lifetime of an isotope against neutron capture. If the cross-section for neutron capture is independent of energy, then the lifetime of the species will be τ n = 1 N n σv 1 N n σ v T = 1 N n σ ( μ n 2kT ) 1/2

Transcript of Heavy Element Nucleosynthesispersonal.psu.edu/rbc3/A534/lec27.pdf · 2017. 6. 26. · Heavy Element...

Page 1: Heavy Element Nucleosynthesispersonal.psu.edu/rbc3/A534/lec27.pdf · 2017. 6. 26. · Heavy Element Nucleosynthesis A summary of the nucleosynthesis of light elements is as follows

Heavy Element Nucleosynthesis

A summary of the nucleosynthesis of light elements is as follows

4He Hydrogen burning3He Incomplete PP chain (H burning)2H, Li, Be, B Non-thermal processes (spallation)14N, 13C, 15N, 17O CNO processing12C, 16O Helium burning18O, 22Ne α captures on 14N (He burning)20Ne, Na, Mg, Al, 28Si Partly from carbon burning

Mg, Al, Si, P, S Partly from oxygen burning

Ar, Ca, Ti, Cr, Fe, Ni Partly from silicon burning

Isotopes heavier than iron (as well as some intermediate weight iso-topes) are made through neutron captures. Recall that the prob-ability for a non-resonant reaction contained two components: anexponential reflective of the quantum tunneling needed to overcomeelectrostatic repulsion, and an inverse energy dependence arisingfrom the de Broglie wavelength of the particles. For neutron cap-tures, there is no electrostatic repulsion, and, in complex nuclei,virtually all particle encounters involve resonances. As a result,neutron capture cross-sections are large, and are very nearly inde-pendent of energy.

To appreciate how heavy elements can be built up, we must firstconsider the lifetime of an isotope against neutron capture. If thecross-section for neutron capture is independent of energy, then thelifetime of the species will be

τn =1

Nn⟨σv⟩≈ 1

Nn⟨σ⟩vT=

1

Nn⟨σ⟩

( µn

2kT

)1/2

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For a typical neutron cross-section of ⟨σ⟩ ∼ 10−25 cm2 and a tem-perature of 5× 108 K, τn ∼ 109/Nn years.

Next consider the stability of a neutron rich isotope. If the ratio ofof neutrons to protons in an atomic nucleus becomes too large, thenucleus becomes unstable to beta-decay, and a neutron is changedinto a proton via

(Z,A+1) −→ (Z+1, A+1) + e− + ν̄e (27.1)

The timescale for this decay is typically on the order of hours, or∼ 10−3 years (with a factor of ∼ 103 scatter).

Now consider an environment where the neutron density is low(Nn ∼ 105 cm−3), so that the lifetime of isotope A,Z against neu-tron capture is long ∼ 104 years. This happens during the heliumthermal pulses of an AGB star, where, due to mixing of shell mate-rial, the reactions

13C+ 4He −→ 16O+ n

14N+ 4He −→ 18O+ 4He −→ 22Ne + 4He −→ 25Mg+ n

release free neutrons at a slow rate. In time, the isotope will capturea neutron, undergoing the reaction

(Z,A) + n −→ (Z,A+1) + γ

If isotope Z,A+1 is stable, then the process will repeat, and after awhile, isotope Z,A+2 will be created. However, if isotope Z,A+1 isunstable to beta-decay, the decay will occur before the next neutroncapture and the result will be isotope Z+1, A+1. This is the s-process; it occurs when the timescale for neutron capture is longerthan the timescale for beta-decay.

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Next, consider a different environment, where the neutron densityis high, Nn ∼ 1023 cm−3. (This occurs during a supernova explo-sion.) At these neutron densities, the timescale for neutron captureis of the order of a millisecond, and isotope Z,A+1 will becomeZ,A+2 via a neutron capture before it can beta-decay. IsotopeZ,A+2 (or its daughter isotope if it is radioactive) would thereforebe an r-process element; its formation requires the rapid capture ofa neutron.

Whether an isotope is an s-process or r-process element (or both)depends on both its properties and the properties of the isotopessurrounding it. In the example above, suppose the radioactive iso-tope Z−3, A+1 is formed during a phase of rapid neutron bom-bardment, and suppose that its daughters Z−2, A+1, Z−1, A+1,and Z,A+1 are all radioactive as well. Isotope Z+1, A+1 wouldthen be the end product of the r-process, as well as the product ofs-process capture and beta-decay from Z,A. On the other hand, ifisotope Z−1, A+1 is stable, then the beta-decays from Z−3, A+1will stop at Z−1, A+1, and element Z+1, A+1 would effectivelybe “shielded” from the r-process. In this case, element Z+1, A+1would be an s-process isotope only. Isotope Z−1, A+1 would, ofcourse, be an r-process element, and, if isotope Z−2, A were unsta-ble, it would be “shielded” from the s-process.

A third process for heavy element formation is the p-process. Theseisotopes are proton-rich and cannot be formed via neutron captureon any timescale. P-process isotopes exist to the left of the “valleyof beta stability” and thus have no radioactive parent. These iso-topes are rare, since their formation requires overcoming a coulombbarrier. Their formation probably occurs in an environment similarto that of the r-process, i.e., where the proton density is extremelyhigh, so that proton captures can occur faster than positron decays.

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A characterization of a portion of the chart of nuclides showingnuclei to the classes of s, r, and p. The s-process path of (n, γ)reactions followed by quick β-decays enters at the lower left andpasses through each nucleus designated by the letter s. Neutron-rich matter undergoes a chain of β-decays terminating at the mostneutron-rich (but stable) isobars; these nuclei are designated by theletter r. The s-process nuclei that are shielded from the r-processare labeled s only. The rare proton-rich nuclei, which are bypassedby both neutron processes are designated by the letter p.

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The valley of stability.

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Isotopic Ratios

The abundance of every isotope A,Z can therefore be written

N(Z,A) = Ns(Z,A) +Nr(Z,A) +Np(Z,A) ≈ Ns(Z,A) +Nr(Z,A)

By knowing the neutron capture cross-sections of the isotopes, andby knowing which isotopes are shielded, (i.e., which are solely s-process and which are solely r-process) the pattern of elementalabundances can be used to trace the history of all the species.

Clearly, the relative abundance of a given s-process element dependson its neutron capture cross-section. If the capture cross-section islow, then nuclei from the previous species will tend to “pile-up”at the element, and abundance of the isotope will be high. Con-versely, if the isotope has a large neutron-capture cross-section, itwill quickly be destroyed and will have a small abundance. S-processabundances can therefore be derived from differential equations ofthe form

dNA

dt= −σANA + σA−1NA−1

The starting point for these equations is 56Ni (although neutroncaptures actually start during helium burning, since the reactions13C(α, n)16O, 17O(α, n)20Ne, 21Ne(α, n)24Mg and 22Ne(α, n)25Mgall create free neutrons). The endpoint of the s-process is 209Bi,since the next element in the series, 210Bi α-decays back to 206Pb.Thus, the entire network looks like

dN56

dt= −σ56N56

dNA

dt= −σANA + σA−1NA−1 57 ≤ A ≤ 209;A ̸= 206

dN206

dt= −σ206N206 + σ205N205 + σ209N209 (27.2)

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with the boundary conditions

NA(t = 0) ={N56(t = 0) A = 560 A > 56

These equations can be simplified locally by noting that the neutroncapture cross-sections for adjacent non-closed-shell nuclei are largeand of the same order of magnitude, so that

σANA ≈ σA−1NA−1 (27.3)

This is called the local approximation; it is only good for some ad-jacent nuclei. Over a length of A ∼ 100, abundances can drop by afactor of ∼ 10.

The abundance of r-process isotopic ratios in nature can be esti-mated by differencing the observed cosmic abundances from thatpredicted from the s-process. Part of the r-process isotopic ratiosmay be due to processes associated with nuclear statistical equilib-rium under extremely high densities (ρ > 1010 gm-cm−3. (Underthese conditions, the Fermi exclusion will force electron captures inthe nucleus, and cause a build up a neutrons.) However, r-processisotopes can also be built up in more moderate densities, if theneutron density is high (∼ 1023 cm−3). Under these conditions, anucleus will continue to absorb neutrons until it can capture no more(i.e., until another neutron would exceed the binding energy). Atthis point, the nucleus must wait until a beta-decay occurs. Thus,the abundance of an isotope with charge Z will be governed by thebeta-decay rates, i.e.,

dNZ

dt= λZ−1NZ−1(t)− λZNZ(t)

where λZ is the beta-decay rate at the waiting point for chargeZ. (In practice, under these circumstances, one must also take into

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account proton and α-particle captures as well, since these reactionswill also add charge to neutron-rich material. However, because ofthe electrostatic repulsion, these capture cross-sections will be muchlower.) R-process nucleosynthesis stops at Z = 94 (N = 175) wherenuclear fission occurs to return two additional seeds to the capturechain.

Measured and estimated neutron-capture cross sections of nuclei onthe s-process path. The neutron energy is near 25 keV. The crosssections show a strong odd-even effect, reflecting average level den-sities in the compound nucleus. Even more obvious is the stronginfluence of closed nuclear shells, or magic numbers, which are asso-ciated with precipitous drops in the cross section. Nucleosynthesisof the s-process is dominated by the small cross sections of theneutron-magic nuclei.

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The solar system σNs curve, i.e., the product of the neutron capturecross sections for kT = 30 keV times the nuclide abundance per106 silicon atoms. The solid curve is the calculated result of anexponential distribution of neutron exposures.

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Calculated r-process abundances after long times. The nuclei havebeen driven around a fission cycle until the abundance distributionbecomes characteristic of a steady state. The abundance of eachnucleus grows exponentially with an efolding time of 4.9 sec. Thistime, calculated on the basis of β-decays rates, depends upon theneutron density and the the temperature.

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The abundances of the elements in the solar system. The dotsrepresent values obtained from the strengths of solar absorptionlines, while the line is based on chemical evidence from the earthand meteorites.

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Nucleocosmochronology

It is possible to obtain an independent estimate of the age of theSolar System (and, indeed the universe) from the study of radioac-tive species (and sometimes their daughter elements). The idea issimple. Consider an radioactive species, i present in an object thatwas born at time τ = t in the universe. Prior to time t, this elementwas created in various events, and had a production rate Pip(t).The abundance of that species today (at time t = t0) will be

N = Pie−λi(t0−t)

∫ t

0

p(τ) e−λi(t−τ)dτ

which simplifies to

N = Pie−λit0

∫ t

0

p(τ) eλiτdτ (27.4)

where λ is the decay rate of the species. Obviously, calculations ofthe production rate can be treacherous. However, if one takes theratio of two elements which are presumed to be created together(say, in r-process events), then most of the unknowns cancel out.So

Ni

Ns=

Pi

Ps

e−λit0

e−λst0

∫ t

0p(τ) eλiτdτ∫ t

0p(τ) eλsτdτ

Moreover, if the second element is stable (with λs = 0), then

Ni

Ns=

Pi

Pse−λit0

∫ t

0p(τ) eλiτdτ∫ t

0p(τ) dτ

≈ Pi

Ps(1− λi(t0 − ⟨t⟩)) (27.5)

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In other words, by observing the ratio of two species, and knowingthe ratio of their production ratio in a star, one can obtain t0 − ⟨t⟩,the mean age of the elements.

Long-lived Galactic Chronometers

Nucleus Half-Life (yrs) Daughter Decay Mode Process

40K 1.3× 109 40Ca β− Si burning40K 1.3× 109 40Ar β+ Si burning87Rb 4.9× 1010 87Sr β− r/s-process138La 1.1× 1011 138Ba β− cap p-process138La 1.1× 1011 138Ce β− p-process147Sm 1.1× 1011 143Nd α s/r-process176Lu 3.7× 1010 176Hf β− s-process187Re 4.5× 1010 187Os β− r-process232Th 1.4× 1010 205Pb Chain r-process235U 7.0× 108 207Pb Chain r-process238U 4.5× 109 206Pb Chain . . .107Pd 6.5× 106 107Ag β− s/r-process129I 1.6× 107 129Xe β− r-process182Hf 9.0× 106 182W β− r-process244Pu 8.2× 107 232Th Chain r-process

Obviously to do this experiment, one needs to work with elementsthat are formed in the same event. In the solar system, one can re-strict the experiment to r-process only isotopes. However, it is notpossible to separate out the different isotopes via stellar absorp-tion lines. (This is not strictly true for the lightest species such asdeuterium and CNO, but for heavier elements it is certainly true.)However, elements heavier than 209Bi can only be made with ther-process (since 210Bi α-decays back to 206Pb).

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The two most popular elements to perform this experiment withis Uranium and 232Th (half-life 13.9 Gyr) and 238U (half-life 4.51Gyr). But these lines are weak!) For example, here are the resultsfrom the extreme metal-poor halo star BD+17 3248 (from Cowanet al. 2002):

Chronometric Age Estimates for BD+17◦ 3248

Element Age (Gyr)Pair Predicted Observed Solar Best Limit

Th/Eu 0.507 0.309 0.4614 10.0 > 8.2Th/Ir 0.0909 0.03113 0.0646 21.7 > 14.8Th/Pt 0.0234 0.0141 0.0323 10.3 > 16.8Th/U 1.805 7.413 2.32 > 13.4 > 11.0U/Ir 0.05036 0.0045 0.0369 > 15.5 > 13.5U/Pt 0.013 0.0019 0.01846 > 12.4 > 14.6

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Fig. 8.—Observed Keck/HIRES and synthetic spectra in regionsurrounding the Th ii !4019.12 line, in the manner of Fig. 3. In the bot-tom panel, the synthetic spectra were computed for abundanceslog "ðThÞ ¼ $1,$1.58,$1.18, and$0.78.

Fig. 9.—Spectra of BD +17%3248 in region surrounding the U ii

!3859.57 line. In both panels the observed Keck/HIRES spectrum is com-pared to four synthetic spectra computed for abundances log "ðUÞ ¼ $1,$2.60, $2.10, and $1.60. (Note that some of the lines overlap.) The onlydifference between the two panels is the vertical scale, which is expanded inthe bottom panel for easier assessment of the extremely weakU ii feature.