reaction rates

Reaction Rates

Marialuisa Aliotta

School of Physics University of Edinburgh

principles of stellar structure and evolution

general features of thermonuclear reactions

experimental approach

Second European Summer School on Experimental Nuclear Astrophysics

St. Tecla, Sept. 28th – Oct. 5th 2003

Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28 th – Oct. 5th 2003

Hertzsprung-Russel (HR) Diagram

Temperature [K]

Lum

inosi

ty

MAINSEQUENCE

WHITE DWARFS

GIANTS

SUPERGIANTS

sun

The Macro-cosmos: some observables

~ 95% of all stars in MAIN SEQUENCE

highest probability of observing them in this stage

(cf. adulthood for human beings)

No chaos, but order!

Surprise!

Stefan´s law: L = 4R2T4

4

o

2

oo TT

RR

LL

Luminosity vs. surface temperature

L ~ M4

more massive stars evolve more rapidly


Mass-Luminosity relationship

(for main sequence stars only)

mass (Msun) lifetime (years)

1 ~1010 5 ~108 10 ~107

L, T, M cannot take up ANY values

ORDER!



Features:

• distribution everywhere similar• 12 orders-of-magnitude span• H ~ 75%• He ~ 23%• C U ~ 2% (“metals”)• D, Li, Be, B under-abundant• exponential decrease up to Fe• peak near Fe• almost flat distribution beyond Fe

Features:

• distribution everywhere similar• 12 orders-of-magnitude span• H ~ 75%• He ~ 23%• C U ~ 2% (“metals”)• D, Li, Be, B under-abundant• exponential decrease up to Fe• peak near Fe• almost flat distribution beyond Fe

Data sources: Earth, Moon, meteorites, stellar (Sun) spectra, cosmic rays...

Abundance curve of the elements



(EXPERIMENTAL) NUCLEAR ASTROPHYSICS

• What is the origin of the elements?• How do stars and galaxies form and evolve?• What powers the stars? • How old is the universe?• …

study energy generation processes in stars

study nucleosynthesis of the elements

NUCLEAR PHYSICS

KEY for understanding

Courtesy: M. Arnould

MACRO-COSMOS

intimately related to

MICRO-COSMOS

Experimental nuclear astrophysics


Quiescent stages of stellar evolution


Stellar structure and evolution controlled by:

1) Gravity collapse

2) Internal pressure expansion

Star composed of many particles (~1057 in the Sun)

Total energy: a) mutual gravitational energy of particles ()

b) internal (kinetic) energy of particles (including photons) (U)

For an ideal gas in hydrostatic equilibrium:

2U + = 0 virial theorem

Assume pressure imbalance

gravitational contraction sets in

amount of energy released -

internal energy change to restore equilibrium U = - ½

gas temperature increases

energy excess - ½ lost from star in form of radiation

Principles of stellar structure and evolution: quiescent evolution


Here: T ~ 10 – 15 X106 K and ~ 102 gcm-3 are required

4H 4He + 2+ + 2 + 26 MeV

ash of nuclear burning

energy source

HYDROGEN BURNING (1st equilibrium)

gravitational contraction of gas (mainly H) increase of central temperature

if T high enough “nuclear burning” takes place

M > 0.1 M (Jupiter = failed star)

gravitational collapse is halted star undergoes phase of hydrostatic equilibrium

MAIN SEQUENCE STARS



Hydrogen burning

Two main mechanisms: proton-proton chain and CNO cycle

M < 1.5 M T6 < 30

p-p chain

M 1.5 M T6 > 30

CNO cycle

(also depends on CNO abundance)

Energy production rate


(Fiorentini’s lecture)


H-burning shellH exhausted in core

isothermal He core

contraction sets in

temperature increases

contracting coreexpanding envelope

RED GIANT STARS

when T ~ 108 K and ~ 103 gcm-3 (minimum mass ~ 0.5 M)

HELIUM BURNING (2nd equilibrium)

3 12C 12C16O

+ 8 MeV

nuclear burning ashes energy source

Wien’s law: maxT = const.

R ~ 10-100 Ri Ts ~ 3-4x103 K



12C/16O BURNING … 12C ashes = Ne, Na, Mg… 16O ashes = Al, … Si

major ash = 28Si SUPER RED-GIANT STARS

28Si MELTING … A = 40-65

major ash = 56Fe PRE-SUPERNOVA STARS

T,

further reactionsbecome endothermic

final gravitational collapse

SUPERNOVA EXPLOSION (type II)

remnant: neutron star or black hole

M 8 M



Stage reached Timescale Tcore (109 K) Density (g cm-

3)

H burning 7x106 y 0.06 5

He burning 5x105 y 0.23 7x102

C/O burning 600 y / 6 months 0.93 – 2.3 2x105 – 1x107

Si melting 1 d 4.1 3x107

Explosive burning 0.1 – 1 s 1.2 - 7 varies

Stellar mass (M) Stage reached

< 0.08no thermonuclear fusion

0.1 -0.5 H burning

0.5 - 8 He burning

8 - 11 C burning

> 11 all stages

Evolution stages of a 25 M star

Principles of stellar structure and evolution: summary


Main parameters: 1) initial mass ( central temperature)

2) initial chemical composition ( nuclear processes)

Quiescent burning

MainSequence~ 1010 y

Red Giant ~ 3x108 y

Super Giant ~ 3x104 y

Pre-Supernova

GravitationalContraction

H

HeC

O, Ne …

e.g. 1/10 M for H-burningless for subsequent stages

Energy generation rate

~ Tn

n ~ 4 (H-burning) n ~ 30 (C-burning)

innermost regions only contribute to nuclear burning

H-burning MAIN SEQUENCElongest stage of star’s lifetime



Explosive stages of stellar evolution


NOVAE

semi-detached binary system:

H-rich mass transfer from RG to WD

temperature and densityincrease on WD’s surface

T > 108 K > 103 g

cm-3

thermonuclear runaway cataclysmic explosion

(p,) and (,p) reactions on proton-rich nuclei

determine nature of nova phenomenon

nucleosynthesis up to A ~ 60 mass region

White Dwarf + less evolved star (e.g. Red Giant)

= sudden increase in star’s luminosity (L ~ 104 – 106 Li and t ~ 1 h – 1 d)

Principles of stellar structure and evolution: explosive evolution

degenerate matter P and T uncoupled


X-RAY BURSTERS & X-RAY PULSARS

semi-detached binary system: Neutron star + less evolved star

T ~ 109 K ~ 106 g cm-3

intense X-ray fluxes

(,p) and (p,) reactions on proton-rich nuclei

nucleosynthesis up to A ~ 80-100 mass region

CORE-COLLAPSE SUPERNOVAE

end stage of M ~ 8-30 M stars

core collapse & rebound shock wave outer layers blown offNeutron Star or Black Hole remnants

T > 109 K

n > 1020 g cm-3

“seed” nuclei in Fe region

(n,) reactions on neutron-rich nuclei followed by decays

nucleosynthesis of n-rich elements through r-process

Principles of stellar structure and evolution: explosive evolution


Stellar life cycle

energy production stability against collapse synthesis of “metals”

thermonuclear reactions

BIRTHgravitational contraction

explosion DEATH

mixing of interstellar gas

Interstellar gas Stars

abundance distribution


Thermonuclear reactions in stars


Thermonuclear reactions in stars: properties of nuclei

Aston: measurements of atomic masses Mnucl < mp + mn E = Mnc2

enormous energy stored in nuclei!

Rutherford (1919): discovery of nuclear reactions

liberate nuclear energy source complex nuclides formed through reactions

Q > 0 Q < 0

Binding energy curve

Q =[(m1+m2)-(m3+m4)]c2 > 0

FUSION reactions most effective in stars

H most abundant element in the Universe

amount of energy liberated in nuclear reaction:

fusion up to Fe region

fission of heavy nuclei

Q > 0

spontaneous nuclear processes:


Consider reaction: 1 + 2 3 + 4 Q12 > 0

Thermonuclear reactions in stars: general features & definitions

( known from atomic mass tables)

Reaction cross section probability for a reaction to occurDimension: area Unit: barn (b) = 10-24 cm2

cross sections depend on nature of force involved

cross sections are energy (i.e. velocity) dependent

Reaction Force (barn) Eproj (MeV)

15N(p,)12C strong 0.5 2.0

3He(,)7Be electromagnetic

10-6 2.0

p(p,e+)d weak 10-20 2.0

In general: not possible to determine reaction cross section from first principles

However:

r =Reaction rate: v(v)N1N2



Maxwell-Boltzmann distribution

Quiescent stellar burning:non-relativistic, non-degenerate gas in thermodynamic equilibrium at temperature T

= reduced massv = relative velocity

kTE

2kTμv2

(v) exp = exp

Pro

babili

ty (

E)

EnergykT

(E) E

(E) exp(-E/kT)

<v>12 =

0

(E) exp E dE

kTE

3/2

1/2

12 kT

1πμ8

0

(v) velocity distribution <v>12 =Reaction rate per particle pair: v(v)(v)dv

In stellar plasma: velocity of particles varies over wide range


Ni = number densityTotal reaction rate: R12 = (1+12)-1 N1N2 <v>12 reactions cm-3 s-1

Energy production rate: 12 = R12 Q12


to be determined from experiments and/or theoretical considerations

as star evolves, T changes evaluate <v> for each temperature

Mean lifetime of nuclei X against destruction by nuclei a

<v> = KEY quantity

energy productionas star evolves

change in abundanceof nuclei X

NEED ANAYLITICAL EXPRESSION FOR !

vN1

)X(a

a =


Non-resonant process

Thermonuclear reactions in stars: reaction mechanisms

Resonant process

Consider reaction: a + X b + Y

One-step process leading to final nucleus Y |<b+Y lHl a+X>|2

single matrix element

occurs at all interaction energies cross section has WEAK energy dependence

Two-step process: 1) compound nucleus formation a + X C*

2) decay of compound nucleus C* b + Y

(b = particle or photon)

|<b+Y lH’l C*>|2 |<C* lHl a+X>|2

two matrix elements

occurs at specific energies cross section has STRONG energy dependence

V

rr0

E

incident nucleusEr

0

Er+1

E1

E2


Reactions between charged particles

nucleosynthesis up to

Fe

typically quiescent

stages


charged particles Coulomb barrier

tunneleffect

Ekin ~ kT (keV) Ecoul ~ Z1Z2 (MeV)

nuclear well

Coulomb potentialV

rr0

determines exponential drop in abundance curve!

in numerical units:

2 = 31.29 Z1Z2(/E)½

in amu and Ecm in keV

kT ~ 8.6 x 10-8 T[K] keV

T ~ 15x106 K (e.g. our Sun) kT ~ 1

keV

T ~ 1010 K (Big Bang) kT ~ 2

MeV

energy available: from thermal motion

Thermonuclear reactions in stars: charged particles

2 = GAMOW factor

reactions occur through TUNNEL

EFFECT

tunneling probability P exp(-2)

during quiescent burnings: kT << Ec

If angular momentum is non zero centrifugal barrier must also be taken into account

(E) = exp(-2) S(E) E1


Non-resonant reactions

2

2

r2)1(V

Thermonuclear reactions in stars: non-resonant reactions

geometrical factor (particle’s de Broglie wavelength)

mEh

ph

2

interaction matrix elementpenetrability probability

depends on projectile’sangular momentum and energy E

22 XaHYb)E(P

Above relation defines ASTROPHYSICAL S(E)-FACTOR

(for s-waves only!)

non-nuclear originSTRONG energy

dependence

nuclear originWEAK energydependence

N.B.


With above definition of cross section:

<v>12 =

0

S(E) exp dE 3/2

1/2

kT1

πμ8

12

1/2Eb

kTE

governs energy dependence

MAXIMUM reaction rate:

Gamow peak

tunnelling throughCoulomb barrier exp(- )

Maxwell-Boltzmanndistribution exp(-E/kT)

rela

tive p

robabili

ty

energykT E0

E/EG

E0

Thermonuclear reactions in stars: Gamow peak

f(E)

0dE

)E(df

varies smoothly with energy

3/2

0 2bkT

E

only small energy range contributes to reaction rate

OK to set S(E) ~ S(E0) = const.

221 eZZ

2b

E0 < E0


reactionCoulomb

barrier (MeV)E0

(keV)exp(-3E0/kT) E0

p + p 0.55 5.9 7.0x10-6

+ 12C 3.43 56 5.9x10-56

16O + 16O 14.07 237 2.5x10-237

Examples: T ~ 15x106 K (T6 = 15)

separate stages: H-burningHe-burningC/O-burning

…

area of Gamow peak (height x width) ~ <v>

E0 = f(Z1, Z2, T)

STRONG sensitivity

to Coulomb barrier

varies depending on reaction and/or temperature

most effective energy region for thermonuclear reactions Gamow peak:

energy window of astrophysical interest E0 ± E0/2

Thermonuclear reactions in stars: Gamow peak


Resonant reactions

(E)BW = π2(1+12) )1J2)(1J2(1J2

21 a b

(E-Er)2 + (/2)2

Breit-Wigner formula

1. Narrow resonances << ER

<v>12 = exp

kTER R

2

3/2

12kTμ2

resonance strength(integrated cross section over resonant region)

insert in expression for reaction rate, integrate and get:

(for single resonance)

Experiment: determine R

and ER

low-energy resonances (ER kT) dominate reaction rate

Thermonuclear reactions in stars: resonant reactions


2. Broad resonances ~ ER

Breit-Wigner formula +

energy dependence of partial a(E), b(E) and total (E) widths

N.B. Overlapping broad resonances of same Jπ interference effects



vtot = vr + vnr

3. Sub-threshold resonances

cross section can be entirely dominated by contribution of sub-threshold state(s)

~ h/

any exited state has a finite width

high energy wing can extend above particle threshold

TOTAL REACTION RATE


Example: 12C(,)16O(Gialanella’s lecture & Schürmann’s talk)


Reactions with neutrons

nucleosynthesis beyond Fe

typically explosive stages


NO Coulomb barrier

neutrons produced in stars are quickly thermalised

E0 ~ kT = relevant energy (e.g. T ~ 1-6x108 K E0 ~ 30

keV)

Typically: v1

~σ <v> ~ const = <TvT>

neutron-capture cross sections can be measured DIRECTLY at relevant energies

accounts for almost flat abundance distribution beyond iron peak

Thermonuclear reactions in stars: neutron captures


Experimental approach

&

laboratory requirements


T ~ 106 - 108 K E0 ~ 100 keV << Ecoul tunnel effect

10-18 barn < < 10-9 barn

average interaction time ~ <v>-1 ~ 109 y

unstable species DO NOT play significant role

Quiescent burning stages of stellar evolution

FEATURES

PROBLEMS 10-18 b < < 10-9 b poor signal-to-noise ratio major experimental challenge extrapolation procedure required

REQUIREMENTS poor signal-to-noise ratio long measurements ultra pure targets high beam intensities high detection efficiency

Experimental approach: general features


measure (E) over as wide a range as possible,

then EXTRAPOLATE down to Gamow energy region around E0!

CROSS SECTION

Experimental procedure

LOGSCALE

direct measurements

E0 Ecoul

Coulomb barrier

(E)

non-resonant

resonance

extrapolation needed !

many orders of magnitude

Experimental approach: extrapolation


Thermonuclear reactions in stars: non resonant reactions

cross section

S-factor

Example:

Data EXTRAPOLATION down to astrophysical energies REQUIRED!


Er

DANGER OF EXTRAPOLATION !

non resonant process

interaction energy E

extrapolationdirect measurement

0

S(E)

LINEARSCALE

S(E)-FACTOR

-Er

sub-threshold resonance

low-energy tailof broad

resonance



ALTERNATIVE SOLUTIONS

Go UNDERGROUND reduce (cosmic) background

example: LUNA facility (Junker’s lecture)

Use INDIRECT methods (Figuera’s lecture)

INTRINSIC LIMITATION

At lower and lower energies

ELECTRON SCREENING EFFECT sets in



The electron screening

in terrestrial laboratories:

interaction between ions (projectiles) and atoms or molecules (target)


(E) = S(E) exp(-2) E1

penetration through Coulomb barrier between BARE nuclei

Rn Rt

Coulo

mb

pote

nti

al

Ec

0

E

bare

screenedE + Ue

RD

in stellar plasmas: ions in sea of free electrons

Debye-Hückel radius

RD ~ (kT/)½

Ue = electron screening potential

Experimental approach: electron screening

Similarly:


fplasma(E) = exp(Ue/E) 1plasma(E)

bare(E)

(E)

screened

bare

E

cross-section enhancement factor:

Experimental approach: electron screening

need to understand flab(E) improve calculation of fplasma(E)

BUT: electron screening in lab DIFFERENT from electron screening in plasma

E0

bare S(E)

S(E)

high-energy dataextrapolation

screened S(E)

fit to measuredlow-energy data

Ue

PROBLEM: experimental Ue >> theoretical Ue


T > 108 K E0 ~ 1 MeV ~ Ecoul

10-6 barn < < 10-3 barn

NO extrapolation needed

average interaction time ~ <v>-1 ~ seconds

unstable species DO GOVERN nuclear processes

Explosive burning stages of stellar evolution

FEATURES

PROBLEMS ~ <v>-1 ~ seconds unknown nuclear properties low beam intensities

(5-10 o.d.m. lower than for stable beams) beam-induced background

REQUIREMENTS unstable species RIBs production and acceleration large area detectors high detection efficiency

Experimental approach: general features


NUCLEAR DATA NEEDS

reactions involving:

A < 30

A > 30

cross-section dependence:

individual resonancesnuclear properties

statistical properties Hauser-Feshbach calculations

excitation energies

spin-parity & widths

decay modes

masses

level densities

part. separation energy

knowledgerequired:

experimental constraints wherever possible

Experimental approach: data needs


EXAMPLES

synthesis of proton-rich nucleiA ~ 100

synthesis of neutron-rich nucleiA > 60

stable

unstable

- decay

+ decay

protoncapture

neutroncapture

rp-process r-process

rapid proton captures X(p,)Y rapid neutron captures X(n,)Y

Z

N

(Shotter’s lecture)

Experimental approach: explosive nucleosynthesis


M.S. Smith and K.E. Rehm, Ann. Rev. Nucl. Part. Sci, 51 (2001) 91-130

Overview of main astrophysical processes


stellar reactions take place through TUNNEL effect

Hydrostatic equilibrium T ~ 106 - 108 K average interaction time ~ <v>-1 ~ 109 y

unstable species do not play significant role

Explosive phenomena T > 108 K average interaction time ~ <v>-1 ~ seconds

unstable nuclei govern nuclear reaction processes

kT << E0 << Ecoul 10-18 barn < < 10-9 barn

Extrapolation NEEDED

Solutions: underground measurements indirect approaches

BUT! Electron screening problem

E0 ~ Ecoul 10-6 barn < < 10-3 barn

Sophisticated techniques for RIBs production and acceleration

Ad-hoc detection systems required

Thermonuclear reactions in stars: overview

Gamow peak

tunnelling throughCoulomb barrier exp(- )

Maxwell-Boltzmanndistribution exp(-E/kT)

rela

tive p

robabili

ty

energykT E0

E/EG

E0


W.D. Arnett and J.W. Truran Nucleosynthesis The University of Chicago Press, 1968

J. Audouze and S. Vauclair An introduction to Nuclear Astrophysics D. Reidel Publishing Company, Dordrecth, 1980

E. Böhm-Vitense Introduction to Stellar Astrophysics, vol. 3 Cambridge University Press, 1992

D.D. Clayton Principles of stellar evolution and nucleosynthesis The University of Chicago Press, 1983

H. Reeves Stellar evolution and Nucleosynthesis Gordon and Breach Sci. Publ., New York, 1968

C.E. Rolfs and W.S. Rodney Cauldrons in the Cosmos The University of Chicago Press, 1988 (…the “Bible”)

Copies of this lecture at: www.ph.ed.ac.uk/~maliotta/teaching

Further reading

reaction rates

Documents

internal energy

inamount of energy

suntotal energy

main sequence stars

massive stars

elementsthe macrocosmos

equilibrium u

ideal gas