the astrophysical formation of the elements

Transcription

1 the astrophysical formation of the elements Rebecca Surman Union College Second Uio-MSU-ORNL-UT School on Topics in Nuclear Physics 3-7 January 2011

2 the astrophysical formation of the elements Chart of the Nuclides proton number Z 4 He mass number Z + N = A neutron number N

3 the astrophysical formation of the elements solar system isotopic and elemental abundances

4 the astrophysical formation of the elements Chart of the Nuclides proton number Z big bang nucleosynthesis ~3/4 H ~1/4 He some Li, Be, B neutron number N

5 the astrophysical formation of the elements Chart of the Nuclides proton number Z Fusion in Stars, Explosive burning H He He C, O C,O Si Si Fe group neutron number N

6 the astrophysical formation of the elements Chart of the Nuclides proton number Z neutron number N

7 the astrophysical formation of the elements nuclear properties nuclear reaction rates astrophysical conditions nuclear network code evolution of nuclear abundances

8 the astrophysical formation of the elements lecture 1: some preliminaries lecture 2: big bang nucleosynthesis lecture 3: fusion in main sequence stars lecture 4: explosive nucleosynthesis lecture 5: neutron-capture nucleosynthesis

9 some terminology n j number of species j per unit volume W j atomic weight (or molar mass) of species j W j = m j N A, where : m j is the mass of one atom of species j N A is Avogadro's number, ± atoms/mole Recall atomic weights are expressed on a scale where W 12 C m u atomic mass unit m u = m 12 C 12 = W 12 C 12 N A = 1 N A = ± g =12 g/mol.

10 some terminology m j mass of one atom of species j m j = (A j Z j )m n + Z j m p + Z j (1 f )m e B j /c 2 m n neutron mass m p proton mass m e electron mass f ionization fraction (0 for a neutral atom, 1 for full ionization) B j nuclear binding energy (note this expression ignores the (small) contribution of the electron binding energy) Δm atomic mass excess, where Δm (m j A j m u )c 2

11 some terminology n j number of species j per unit volume W j atomic weight (or molar mass) of species j ρ m mass density ρ baryon mass density ρ m = j n j W j N A ρ = n j A j j N A X j nucleon fraction, or mass fraction Y j mole fraction, or abundance X j = n j A j ρn A Note X j = j j n j A j ρn A = 1 ρ j n j A j N A =1 Y j = X j A j = n j ρn A

12 some terminology y j rescaled abundances In meteoritics, abundances are normally scaled relative to silicon (set number of silicon atoms to be 10 6 ) : log y j = log f Si + logy i where f Si is the appropriate normalizing constant. For Y Si = , log f Si = Astronomers sometimes use a scale relative to hydrogen (set number of hydrogen atoms to be ) : log y j = log f H + logy i where log f H = More commonly, astronomers express abundances as ratios relative to solar : log[n i /n j ] star log[n i /n j ] solar [i / j] For example, a ratio of sodium to iron which is half solar would be [Na/Fe] = -0.3.

13 example abundance patterns Letarte et al (2006) log(ε i ) = log 10 ( ) +12 n i /n H Roederer et al (2009)

14 the astrophysical formation of the elements nuclear properties nuclear reaction rates astrophysical conditions nuclear network code evolution of nuclear abundances

15 building a nuclear network code: nuclear properties atomic/nuclear masses How determined? mass spectroscopy

16 building a nuclear network code: nuclear properties atomic/nuclear masses How determined? Penning trap

17 building a nuclear network code: nuclear properties atomic/nuclear masses

18 building a nuclear network code: nuclear properties atomic/nuclear masses How determined? theoretical models Weizs a cker - Bethe semiempirical mass formula B /c 2 = a 1 A a 2 A 2 / 3 a 3 Z 2 A 1/ 3 a 4 A 2Z 2 A ± a 5 A 1/ 2 Figure by MIT OpenCourseWare. From Evans

19 building a nuclear network code: nuclear properties atomic/nuclear masses How determined? theoretical models ex: finite-range droplet model

20 building a nuclear network code: nuclear properties radioactive decay rates N(t) = N 0 e λt λ decay constant τ mean lifetime, τ = 1/λ T 1/ 2 half - life, T 1/ 2 = ln2/λ How determined? MSU/NSCL beta counting system

21 building a nuclear network code: nuclear properties radioactive decay rates N(t) = N 0 e λt λ decay constant τ mean lifetime, τ = 1/λ T 1/ 2 half - life, T 1/ 2 = ln2/λ How determined? Fermi's 'Golden Rule' rate = 2π f H int i 2 ρ(e) f, i final and initial state wavefunctions H int weak interaction Hamiltonian ρ(e) density of states for the final particles

22 building a nuclear network code: reaction rates cross section for the reaction i + j k + l σ ij (v) = number of reactions per nucleus i per second flux of incoming projectiles j σ ij (v) = r ij /n i n j v ij r ij number of interactions i( j,k)l per second v ij relative velocity of particles i, j

23 building a nuclear network code: reaction rates cross section for the reaction i + j k + l σ ij (v) = number of reactions per nucleus i per second flux of incoming projectiles j σ ij (v) = r ij /n i n j v ij r ij number of interactions i( j,k)l per second v ij relative velocity of particles i, j So the reaction rate per unit volume is just: r ij = n i n j v ij σ ij (v)

24 building a nuclear network code: reaction rates In astrophysical environments the relative velocity v ij is not constant, but instead there exists a distribution of relative velocities, which can be described by the probability function P(v), where: P(v)dv =1 So the reaction rate can be generalized to: 0 r ij = n i n j vp(v)σ ij ( v)dv 0 r ij = n i n j σv ij

25 building a nuclear network code: reaction rates If the nuclei are nonrelativistic and nondegenerate, their velocities can be described by a Maxwell-Boltzmann distribution 3 / 2 m P(v)dv = ij e m ij v 2 / 2kT 4πv 2 dv 2πkT where : m ij reduced mass, m ij = m i m j /(m i + m j ) T k temperature Boltzmann constant, k = ev/k The velocity distribution can be written as an energy distribution, since E = m ij v 2 /2 P(v)dv = P(E)dE = 2 π 1 (kt) 3 / 2 Ee E / kt de

26 building a nuclear network code: reaction rates So the reaction rate per particle pair becomes: σv ij = vp(v)σ(v)dv = vp(e)σ(e) de 0 = 8 πm ij 1/ 2 1 (kt) 3 / 2 0 Eσ(E)e E / kt de 0 The cross section σ(e) is often parameterized in terms of the astrophysical S-factor: σ(e) 1 E e 2πη S(E) η = Z iz j α 2E /m ij s-wave Coulomb barrier transmission probability

27 building a nuclear network code: reaction rates Then we have: σv ij = 8 πm ij 1/ 2 1 (kt) 3 / 2 0 e 2πη S(E)e E / kt de Gamow peak Nuclear Physics of Stars, Illiadis (2007)

28 building a nuclear network code: reaction rates A side note: The Gamow peak shifts up in energy with increasing Z i, Z j, and the area under peak decreases Nuclear Physics of Stars, Illiadis (2007) Therefore in stellar plasmas, the fusion of light nuclei is more probable and releases more energy than fusion of heavier nuclei

29 building a nuclear network code: reaction rates Also note: while S(E) is generally an experimentally-determined quantity, extrapolation to lower energies is often required for astrophysical applications

30 building a nuclear network code: reaction rates Often see the S-factor further parameterized (Bahcall, 1989): σv ij = 8 πm ij 1/ 2 1 (kt) 3 / 2 Eσ(E)e E / kt de 0 σ(e) 1 E e 2πη S(E) ( ) Z Z i j σv ij = cm 3 /s where : E 0 /kt = πz i Z j α / 2 A ( ) 2 / 3 m ij /kt [ ] 1/ 3 1/ 3 f 0 S eff T 9 2 / 3 e 3E 0 / kt Gamow peak energy A = A ia j A i + A j S eff = S(0) 1+ 5kT + S (0)E kT E 0 36E 0 2 S (0)E kT 36E 0

31 building a nuclear network code: reaction rates Inverse reactions: i + j k + l Recall we have: σv ij kl = σv kl ij = 8 πm ij 8 πm kl 1/ 2 1/ 2 1 (kt) 3 / 2 1 (kt) 3 / 2 Eσ ij kl e E / kt de 0 Eσ kl ij e E / kt de 0 The ratio of the two is given by: σv kl ij = m ij σv ij kl m kl 3 / 2 (2 j i +1)(2 j j +1) (2 j k +1)(2 j l +1) e Q ij kl / kt where Q ij kl = E kl E ij

32 building a nuclear network code Now consider the rate of change in the number density of species j: dn j dt = n k n l σv kl, j n j n l σv jl,n + n i λ i, j n j λ j,m + Note for reactions involving identical particles, a term of the form: n i 2 2! σv (two body) or n i ii, j 3! σv (three body) iii, j is needed. The above can be written in terms of abundances as: 3 dy j dt = Y k Y l ρn A σv kl, j Y j Y l ρn A σv jl,n + Y i λ i, j Y j λ j,m + this is often what we call the reaction rate in a network code

33 recommended reading Supernovae and Nucleosynthesis, David Arnett, Princeton University, Introduction to Nuclear Astrophysics, Richard N. Boyd, University of Chicago, Principles of Stellar Evolution and Nucleosynthesis, Donald D. Clayton, University of Chicago, Nuclear Physics of Stars, Christian Illiadis, Wiley, 2007.