Topic 5 Neutron capture and the production of heavy elements Elements beyond the Iron peak ! We have already seen that thermonuclear fusion beyond Iron is endothermic ! If thermonuclear burning created the elements beyond Iron we would see a pattern of abundance dropping rapidly with A due to the increasingly large Coulomb barrier ➙ need another process, that of neutron capture ! Free neutrons are unstable particles with a half life of about 10 minutes ! Nuclei formed by neutron capture are often unstable with respect to β decay Evidence for neutron capture 1. At certain stages in a star’s evolution it is known that large fluxes of neutrons are created in the stellar interiors 2. Neutron capture cross-sections for heavy elements are very large compared with those for light elements 3. Only 3% of Iron peak elements are needed to synthesise all A>60 heavy elements ➙ enough “seed material” exists 4. Observation of Technetium in stellar atmospheres: half-life of Tc is ~ 106 yr i.e. less than star’s lifetime ➙ must have been produced in the star (note: no Tc on Earth) Neutron capture processes ! There are, in fact, 2 neutron capture processes: ! The s-process • Neutron capture time is much larger than β-decay lifetime • Neutron energy range is typically 1 to 300 keV • Synthesised elements closely follow the line of β stability • Timescales vary wildly (minutes → 106 years) ! The r-process • Neutron capture times (10-3 to 10-4s) are shorter than β- decay lifetimes • Synthesised elements lie away from the line of β stability Chart of nuclides I Image from http://webassign.net/pas/chart_nuclides Chart of nuclides II Image from http://www.quantum.physik.uni-mainz.de/mats/experiments/precision-ms Zoom-in on Chart of Nuclides Chart of Nuclides Observations: ! Some nuclei can be produced by both r- and s- processes ! Some can only be produced by the s- process, generally they have relatively few neutrons ! Some can only be produced by the r-process, includes all those heavier than lead or bismuth and generally those with the largest neutron excess ! A few nuclei cannot produced by either process, they have the largest number of ! Production of bypassed nuclei: protons. These are known as bypassed ! (p,γ) and (p,n) reactions which could occur nuclei and may have been produced by during explosive burning proton capture. Altogether there are about 30 ! (γ,n) reactions - at temperatures of about 9 nuclei of this type, they are much less 10 K thermal radiation can have sufficient energy to eject neutrons from nuclei abundant ! Inverse beta-decay - at high T results in e+e- ! Note also the even-odd effect due to nuclear pairs in thermal equilibrium, e+ capture by a structure. There are fewer stable nuclides nucleus can produce a more proton rich with odd Z and/or odd N species. The s-process ! S-process neutrons are produced during the Helium burning phase in Red Giants via processes such as: 13C + 4He " 16O + n 16O + 16O → 31S + n and 14N + 4He → 18F 18 18 + F → O + e + νe 18O + 4He → 22Ne 22Ne + 4He → 25Mg + n The s-process ! If nuclei are placed in a neutron flux then the (n, γ) capture process produces isotopes of progressively larger A (Z,A) + n → (Z,A+1) + γ ! When this process creates an unstable nucleus then the path switches to the next Z value via β-decay - (Z,A+1) → (Z+1,A+1) + e + νe then neutron capture continues as before. ! Generally the present day abundance of a particular nuclide will have been the result of both s and r- processes but for those which can be produced only by the s-process we would expect some relationship with neutron capture cross-section up leading to larger abundance. Neutron cross-sections and fluxes ! To a high degree of accuracy, neutron cross-sections σ are inversely proportional to neutron velocity v. This is true throughout the range of velocities relevant to the s- and r- processes ! The product σv can be considered constant. For a heavy nucleus σv ~ 3 x 10-23 m3 s-1 ! For a neutron density n m-3 such a nucleus will capture a neutron in a time t where t = (nσv)-1 = 3 x 1022(1/n) s = 1015(1/n) years ! If the “characteristic” s-process time t = 104 years then neutron density n ~ 1011 m-3 ! Similarly, the r-process requires n ~1022 to 1030 m-3. Implies a significant fraction of local matter is converted into neutrons s-process production ! For s-process production in a neutron density n the rate of production of nucleus A is given by : dNA/dt = n { (σA-1v)NA-1 – (σAv)NA } where the two terms represent production and destruction respectively. ! Since σv is constant use σth(thermally averaged cross- -1 section) and vth (~ 2200 ms ): dNA/dt = vthn{(σA-1)thNA-1 – (σA)thNA} ! Define neutron exposure as τ ≡ ∫ Φdt = ∫ nvth dt to give: dNA/dτ = σA-1NA-1 - σANA ! So s-process is represented by a large number of linked differential equations.€ At any time the value of NA influences the rate of production of neighbouring elements. s-process boundary conditions ! Using these linked differential equations to calculate abundances requires some boundary conditions: ! Since in solar abundances N(53Fe ) ~ N(57Fe) ~ 0.1N(56Fe) the starting material is assumed to be 56Fe ! There is also a natural termination to the s-process sequence since 209Bi is the most massive stable nucleus. Neutron capture by 209Bi leads to a nucleus which decays by alpha particle emission to 206Pb So: dN56/dτ = - σ56N56 dNA/dτ = σA-1NA-1 - σANA dN206/dτ = σ205N205 - σ206N206 + σ209N209 Initial conditions: NA(0) = N(0) at A = 56, NA(0) = 0 for all other A Local approximation ! It follows that: dNA/dτ < 0 if σANA > σA-1NA-1 dNA/dτ > 0 if σANA < σA-1NA-1 Thus the system is self regulating - decreasing NA if it is high and increasing NA if it is low - tending Above: σ vs. N towards the balance σANA ~ σA-1NA-1 Below: Nσ vs. N i.e. dNA/dτ = 0 ! This equality is called the local approximation. It is not expected to work where σ the cross-section is particularly low - for example in the region of the magic numbers of neutrons, N = 50, 82, 126. Local approximation: evidence ! Convincing evidence to support the local approximation can be obtained from looking e.g. at the isotopes of samarium: -31 2 A NA (%) Process σ (10 m ) Nσ 144 2.8 p 119 ± 55 342 ± 158 147 14.94 r,s 1173 ± 192 17600 ± 2900 148 11.24 s 258 ± 48 2930 ± 540 149 13.85 r,s 1622 ± 279 22500 ± 3900 150 7.36 s 370 ± 72 2770 ± 535 152 26.90 r,s 411 ± 71 11100 ± 1900 154 22.84 r 325 ± 61 7400 ± 1400 here (σN)148/(σN)150 = 1.06 ± 0.27 more accurate direct measurements of cross-sections yield (σN)148/(σN)150 = 1.02 ± 0.06 in excellent agreement with the local approximation. The r-process ! The r-process is much less straightforward than the s- process both from the point of view of calculating expected abundances and the supporting evidence from observed abundances ! The major factor supporting the r-process is the existence of elements heavier than 209Bi for the production of which no other explanation is known ! Similarly the existence of neutron-rich isotopes which cannot easily be produced by the s-process (owing to intervening unstable isotopes) supports the model ! To obtain the expected abundances due to the r-process the contribution due to the s- process, calculated using the coupled differential equations, is first subtracted from the observed elemental abundances The r-process Why are these peaks at a lower A value? Stable nuclei are not produced directly by the r-process, only indirectly by β-decay from the neutron rich nuclei. In β-decay neutrons are transformed into protons so A is conserved. Therefore the value of A for, e.g. N=126 is lower in the neutron rich nuclei than it is in the stable nuclei. ! There are strong indications of peaks at A = 80, 130 and 195 which are about ten units below the atomic numbers equivalent to magic numbers of neutrons, i.e: A = 90 (N = 50), 140 (N = 82), 208 (N = 126). The neutron capture cross- section is exceptionally small at these values of N. The r-process and waiting points ! To help understand the r-process we consider nuclear binding energies: 2 2 2 M(A, Z) c = Z mp c + (A - Z) mnc - B(A, Z) where B the binding energy. ! The rapid neutron absorption process M(A, Z) + mnM(A+1, Z) is repeated over and over until the next neutron is not bound – i.e. it does not contribute to the binding energy and so B(A, Z) = B(A+1, Z) ! At this point the r-process stops until a beta decay occurs ! The points (on the chart of nuclides) where the r-process stops are called halt points or waiting points r-process ! In high temperature environments it is possible there is another mechanism to halt the r-process that of photo-nuclear emission due to the thermal photons ! For example for a neutron density of 1030 m-3 and temperature 109 K the process stops at Q(A+1, Z) = B(A+1, Z) - B(A, Z) ~ 2 MeV since the probability of photo-neutron emission is just equal to the probability of neutron capture r-process production ! For each value of Z there will be a waiting point beyond which the process of neutron absorption cannot continue until a beta-decay occurs: ! Thus for the abundance NZ at each Z value we can write: dNZ/dt =λZ-1NZ-1 –λZNZ where λZ is the decay constant at the waiting point for charge Z (note λ is the same as in N=N0exp(-λt)).