Nuclear Astrophysics Lecture 1. L. Buchmann TRIUMF

Transcription

1 Nucer Astrophysics Lecture L. Buchmnn TRIUMF Jnury 06 Kot

2 Eduction is wht remins fter you hve forgotten everything you hve erned. iε Jnury 06 Kot

3 Nucer Astrophysics The uminous structure of the Universe is rgey given by the properties of the nucei inhbiting it. These provide the energy source nd thus the structure of strs nd produce this wy new nucei nd eements. Nucer Physics expins the woring of nucei. Mny properties of nucei nd nucer scttering re nown from experiments with no prticur interest in strophysic questions. Nucer Astrophysics is genery the nucer physic dedicted to questions rising directy from strophysic probems. However, the fied is rther brod nd ony few probems wi be discussed here. E.g. there wi be no discussion of neutron physics, e.g. cpture cross sections, r-process or others. Jnury 06 Kot

4 Overview of ectures. A itte ster stronomy. A bit more on scttering theory st dy 3. C,γ 6 O, some discussion, new resuts C,γ 44 Ti 5. 7 Bep,γ 8 B experiment 6. 7 Bep,p 7 Be nd TACTIC 7. Rdioctive bem experiments t TRIUMF Jnury 06 Kot

5 A Str Rdition Grvity After ong rndom w ight is rdited wy. Rdition Jnury 06 Kot Strs re most of their ives in qusisttic equiibrium between rdition produced in nucer burning nd grvity puing ginst the rditive pressure.

6 Strs twine Jnury 06 Gobur custer M3: RR Lyre strs Kot

7 The recircution of mtter Mtter in strs Gctic couds Gctic couds Nucei re the DNA of the Universe. Stn Woosey Jnury 06 Kot

8 Cssifying strs: The Hertzsprung-Russe Digrm Strs cn be cssified ccording to surfce temperture nd uminosity. Ster evoution eds to pth through the HRD. Ster evoution is directy ined to microscopic, nucer physics. Jnury 06 Kot

9 The microscopic word: prtice scttering Y I n σ θ, Ω dω d dr + m + V r] y r r h [ Prti wve decomposition 0 Jnury 06 Kot

10 Wht re Phseshifts? ψ i z r e + f Ω e r ir Rdiy divergent outgoing wve σ Ω f Ω Cross section squre of scttering mpitude f θ 0 f P cosθ Prti wve decomposition iδ π e sinδ σ 4 δ tot + sin 0 with the phseshift δ Strongy decresing potenti, no Couomb forces, no spins. Jnury 06 Kot f +.

11 Infinite rnge: needs speci tretment. Couomb Potenti Sommerfed prmeter: ZZ V c. [MeV] /3 / 3 A + A η Z Ze hv ZZ m[u] E[MeV] Couomb scttering mpitude: η f C θ exp[ iη nsin θ + iϖ 0] sin θ Jnury 06 Kot

12 dσ θ, E dω + 0 Jnury 06 More soutions to the Couomb probem + P Rdi Schrödinger eqution for Couomb probem: y η exp i[ω η r + ] y r + [ Two soutions: F,,.., G, sin θ + δ ρ η ρ η exp[ iη n sin θ ] E]sinδ η rctn 0 m Spiness prtices, with ω, > 0, ω 0 m Kot E 0 regur nd irregur Couomb functions, ρ r Phse shift nysis

13 Derivtion of the R-function Spiness cse, one chnne, sttes in squre potenti; then set of eigensttes fufiing the Schrödinger eqution: h m d χ dr + V r These cn obey boundry condition t ny mtching rdius: dχ B χ dr + χ An unbound prtice scttering t the potenti hs to so fufi the Schrödinger eqution: h m d φ dr + V r φ 0 E Eφ χ E > 0 E 0 E Jnury 06 Kot

14 Derivtion of the R-function From competeness nd orthonormity foows: φ A χ A 0 χ φdr Mutipying the two Schrödinger equtions with the opposite wvefunction nd subtrction s we s integrtion from 0 to gives: 0 χ d ϕ dr ϕ d χ dr Prti integrtion eds to: dr + m E E h 0 φχ dφ dχ m χ φ r + E E 0 A dr dr h Appying the boundry condition nd resoving eds to: dr 0 ϕ r G r, { ϕ + Bϕ } Jnury 06 Kot

15 Jnury 06 Kot Derivtion of the R-function With the Green s function,, G R E E r m r G χ χ h Defining: γ χ γ E E R m h The ogrithmic derivtive t is then: R BR φ ϕ At the intern wvefunctions re mtched to the extern one: R BR O U I O U I ϕ ϕ ψ ψ

16 Scttering Mtrix With defining the scttering mtrix U: ψ In singe chnne R-mtrix theory is then: I UO I incoming wvefunction, O outgoing. U I O L L * R R The rest Couomb properties: P F I ρ + G, + G if e O G if iω iω L S + ip with S P F F + GG Shift function Penetrbiity e Jnury 06 Kot

17 Jnury 06 Generize the scttering mtrix The scttering mtrix cn be generized for prtices with spins nd rections with mutipe chnnes, so tht for non estic rections is generiztion for estic possibe: With the foowing symbos: σ π J g J U s, s J ss, physic chnne p,, others ; -wve number J tot spins of compound sttes; U scttering mtrix s incoming chnne spin ngur momentum g J spin sttistic fctor Kot i.e., eements of the scttering mtrix re squre roots of reduced cross sections for individu quntum numbers.

18 Wht re Phseshifts? Then, more formy, phseshifts re: U J s, s e i ω + δ J s Jnury 06 ω with being the Couomb phse nd J δ s the nucer phseshift, usuy compex number, re prt estic, imginry prt sum of inestic chnnes.. As fr s estic scttering is concerned Couomb term nd Couomb-nucer interference term hve to be dded. Kot

19 Heium Burning When hydrogen burning hs stopped in the core it strts to contrct sowy ti heium ignites. -> Heium Burning. The process encompsses ony two nucer rections s fr s energy production is concerned. 3 C C,γ 6 O Jnury 06 Kot

20 Importnce of 6 C,γ O SNI pre-exposion deveopment S. Woosey, 00 Jnury 06 Kot

21 Reevnt 6 O stte structure 300 ev corresponds to quiescent heium burning. Jnury 06 Kot

22 Composition of the cross section. Rditive cpture to the ground stte: i, E component. ii, E component.. Cscde trnsitions: i 6.0 stte: structury the sme s gs. ii 6. stte:,,3,4,5 possibe, but itte observed. iii 6.9 stte: 0,,,3,4: 0,,4 nd 3 coherent in tot cross section. The strongest cscde. Jnury 06 iv 7. stte:,,3,4 possibe, itte observed. Kot

23 Wht do we now?. The bsic nucer structure of 6 O.. Mny mesurement of the 6 cross section between MeV cm. C,γ 3. Direct determintion of -widths by scttering methods, ie estics or the 6 N spectrum. 4. γ-decy widths 5. More indirect informtion ie from trnsfer rections. O How to bring it together? Dt, so ced dt, nd theory Jnury 06 Kot

24 phseshifts So ced dt Phseshifts From Notre Dme dt Jnury 06 Kot

25 Jnury 06 Eectromgnetic trnsitions From the Mxwe equtions the chrge nd current free equtions for the vector potenti A cn be derived s: A c A t 0 A Eectric nd mgnetic fieds re given by derivtives of the vector potenti: E LM A E H A c t Looing for sttionry soutions nd seprting by ngur momentum gives two soutions for the eectromgnetic rdition `eectric nd mgnetic : A C Lu A ic L LM Kot M LM L 0 Lu with the soutions of the one dimension Hemhotz eqution: u LM j L LM L ωr / c r YLM θ, φ YLM θ, φ L +!!

26 Jnury 06 Kot Eectromgnetic trnsitions With the tter pproximtion becuse of the ong wveengths of gmm rdition compred to nucer size. First order perturbtion theory gives the trnsition probbiity s: de dn i f T f i > < H h π with the hmitonin H, buit upon the vector potenti A. H A p σ µ mc e mc e p h - H After ots of pproximtions is then: L ML f i L EL f i c mc c e O T c c e O T ω ω ω ω h h h L L L f i L f i M / E / π π π π π π Prity retions:

27 Jnury 06 Some bsics bout poputing sttes ngur moment ddition: Eigenfunctions of c: c + c γb > > bβ >, bβ cγ β Kot b Cebsch-Gordn coeff. 3 vectors: Rch Coefficients W, 4 vectors 9 j symbos {}. Normy, we observe mny, i.e. n ensembe, of toms/nucei being uncorreted described by n impure stte Ψ. An eement of the density mtrix is then < A ρ A > [ < A Ψ >< Ψ A > ] for set of quntum numbers A which describe feture of the system. The expecttion vue of n opertor cting on this system is [ Q ] Tr[ ρ < Ψ Q Ψ > A Q A > [ < A Ψ vg < >< Ψ vg AA ] For the eigensttes > density tensor cn be constructed so tht: ρ κ, κ < ρ > A > ] vg Tr [ Q ρ Exmpe: no poriztion < ρ > δ / ˆ nd ρ 00 / ˆ ]

28 Jnury 06 Kot More bsics bout poputing sttes Unitrity resuts in:, κ κ ρ κ ρ > < Simir the efficiency tensor is: > <, ε κ ε κ Then is the corretion function is: > >< < κ κ κ ε ρ ε ρ ρε Tr W ] [ Wigner-Ecrt theorem: Be the fin product stte from: > > > β β b b β β V b b >< < < Then there is n interction mtrix eement independent of, β, nd so tht, β β b b b < > < <

29 Ground stte ngur distributions s + s b + b c + L b L c Expressions cn be derived from e.g. A.J. Ferguson 965, requiring to wor out ngur corretion coefficients nd eep the ccounting right. For one trnsition the ngur distribution is very gener: W W θ < b ρ >< b κ c b 6π Z Jnury 06 Kot b * * bb ρ ε bb ε > b b κ The density mtrices cn expressed by the initi stte : ρ κ b b bb ρ κ For prtice-γ cscde is then: ˆ * κ < c L b b; Z κ b >< c L * { 9 j} < b >< b > ρ κ... L bl b > * b; c Q P cosθ

30 with with with with Ground stte ngur distributions From ngur moment ddition rues the ground stte 6 ngur distribution in cn be derived s C,γ W θ γ, E W θγ, E + W θγ, E + W σ E + σ W W E O W θ, E P cos θ γ γ θ σ E 5 θγ, E + P cos θγ P4 σ E 7 7 γ, E cos θ γ 6 σ E θ, [5 ] / γ E cos Φ P cos θγ P3 cos θ 5 σ E γ Φ δ δ + rctn η Jnury 06 Kot