example: direct capture

cross sections for direct reactions

)E(PAxHB22

x

A + x B + g

“geometrical factor”

de Broglie wave length

of projectile

mE2

h

p

h

matrix element

contains nuclear

properties of

interaction

- Penetrability/ Transmission

probability of projectile to interact

with target nucleus

- Depends on angular momentum

of projectil and energy E

)E(S)E(PE

1

Needed: penetrability P(E)

Transmission Probability depends on:

Coulomb barrier (only charged particles)

centrifugal barrier (for neutrons and charged particles)

= (strong energy dependence) x (weak energy dependence)

S(E) = astrophysical factor

Contains nuclear physics of reaction, matrix element, wave functions, operator

Reminder: cross section for direct reaction

stellar reaction rates for charged particle capture

EdE)kT/Eexp()E(vdv)v()v(v

dEE

b

kT

Eexp)E(Sv

maximum of reaction rate at E0: 0E

b

kT

Eexp

dE

d

Gamow peak

Tunnel effect

Coulomb barrier

exp(- )

Maxwell-Boltzmann

distribution

exp(-E/kT)

rela

tive

Wa

hrs

che

inlic

hkeit

energykT E0

E/EG

E0

E0 = relevant energy for astrophysics >> kT

3/2

6

3/12

2

2

1

3/2

0 22.12

TZZbkT

E

keV

6/5

6

6/12

2

2

10 749.03

4TZZkTEE keV

Gamow peak

Remark Gamow energy depends on reaction and temperature

and substitution for :

Reminder: Gamow peak

Orbital angular momentum is conserved for central potential

For higher angular momentum transfer the linear momentum p has to be

larger for same d

angular momentum barrier

Angular momentum of incoming particle

z-axis

incoming

particle

target

nucleus

p

dp = projectile momentum

d = impact parameter

classical physics:

L=p x d

quantum mechanics:

)1(L = 0 s - wave

= 1 p - wave

= 2 d - wave

…with parity of wave function: = (-1)

(discrete values)

Orbital angular momentum is conserved for central potential

finite orbital angular momentum implies “orbital angular momentum energy barrier” V

2

2

r2

)1(V

= reduced mass of projectile-target system

r = radial distance from target nucleus center

Angular momentum

neutron capture

Simplest case: s-wave neutrons V = 0 and Coulomb potential VC = 0

attractive nuclear potential

incoming wave transmitted wave

reflected wave

-V0

Discontinuity of potential causes partial reflection of incoming wave

Transmission probability:

2/1EP for = 0

2/1EP for 0

v

1

E

1

2/1E

s wave: neutron capture is dominating usually at

low energies (exception: hinderance due to selection rule)

V=0

higher values: neutron capture is only possible at higher energies

relevant (or when =0 capture is suppressed)

consequences:

Transmission probability

neutron capture

Cross section is strongly reduced

at lower energies

(angular momentum barriere)

dependence of neutron capture

cross section

-0.5 0.0 0.5 1.0 1.5 2.01E-3

0.01

0.1

1

10

= 3

= 2

= 1

cro

ss s

ection [a.u

]

energy [a.u.]

= 02/1E

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.1

1

10

100

= 3

= 2

= 1

penetr

abili

ty [a.u

.]

energy [keV]

= 0

2/1EP

Remark: arbitrary scale between values

low values dominate reaction rate

at low energies

dependence of penetrability

through centrifugal barrier

Angular momentum: pentrability & cross section

stellar reaction rates for neutron capture

v

1

vconstv

s-wave neutron capture

0 10 20 30 40 50 60 70 800

1

2

3

4

(E)F(E)

neutron flux distribution

F(E)

(Maxwell-Boltzmann)

arb

itra

ry u

nits

energy [keV]

(E) = 0

relevant energy region

E ~ kT

th = measured cross section for thermal neutrons

kT2vT

EdEkTEEvdvvvv )/exp()()()(

most probable velocity Ecm = kTthTvv

Stellare Reaktionsrate

neutron capture

example: 7Li(n,g)8Li

Deviation from 1/v behavior due

to resonant contribution

thermal cross section

> = 45.4 mb

s-wave neutron capture

case: = 0

v

1

E

1

cross section for resonant reactions

for a single isolated resonance:

resonant cross section given by Breit-Wigner expression

22

r

21

T1

2

2/EE1J21J2

1J2E

for reaction: 1 + T C F + 2

geometrical factor 1/E

spin factor

J = spin of CN’s state

J1 = spin of projectile

JT = spin of target

strongly energy-dependent term

1 = partial width for decay via emission of particle 1

= probability of compound formation via entrance channel

2 = partial width for decay via emission of particle 2

= probability of compound decay via exit channel

= total width of compound’s excited state

= 1 + 2 + g + …

Er = resonance energy

what about penetrability considerations? look for energy dependence in partial widths!

partial widths are NOT constant but energy dependent!

cross section for resonant reactions

particle widths 211 )E(P

R

2

P gives strong energy dependence

R radius of nuclear potential

= “reduced width” (contains nuclear physics info)

energy dependence of partial widths

example: 16O(p,g)17F

energy dependence of proton

partial width p as function of

particle partial widths have approximately

same energy dependence as penetrability

function seen in direct reaction processes

EdE)kT/Eexp()E(vdv)v()v(v

here Breit-Wigner cross section

22

r

21

T1

2

2/EE1J21J2

1J2E

if compound nucleus has an exited state (or its wing) in this energy range

RESONANT contribution to reaction rate (if allowed by selection rules)

typically:

resonant contribution dominates reaction rate

reaction rate critically depends on resonant state properties

two simplifying cases:

narrow (isolated) resonances

broad resonances

<< ER

narrow resonance case

g

kT

Eexp

kT

2v R

R

2

2/3

12

12

resonance must be near relevant energy range E0 to contribute to stellar rate

MB distribution assumed constant over resonance region

partial widths also constant, i.e. i(E) i(ER)

reaction rate for a single narrow resonance

NOTE

exponential dependence on energy means:

rate strongly dominated by low-energy resonances (ER kT) if any

small uncertainties in ER (even a few keV) imply large uncertainties in reaction rate

reaction rate for:

narrow resonances

resonance strength

g 21

T1 )1J2)(1J2(

1J2

(= integrated cross section

over resonant region)

often 21

121

221

221

112

reaction rate is determined by the smaller width !

i values at resonant energies)

g

kT

Eexp

kT

2v R

R

2

2/3

12

12

some considerations…

rate entirely determined by “resonance strength” g and energy of the resonance ER

experimental info needed:

partial widths i

spin J

energy ER

note: for many unstable nuclei most of

these parameters are UNKNOWN!

)12)(12(

12

1

TJJ

J

21g

statistical factor:

width ratio:

example: 24Mg(p,g)25Al

the cross section

almost constant S-factor

direct capture contribution

non-constant S-factor

resonant contribution

… and the corresponding S-factor

Note varying widths of resonant states !

22R

21

2

)2/()EE()E()E(

assume:

2= const, = const and use simplified

broad resonance case

~ ER

same energy dependence

as in direct process

for E << ER very weak

energy dependence

broader than the relevant energy

window for the given temperature

resonances outside the energy range

can also contribute through their wings

Breit-Wigner formula

+

energy dependence of partial and total widths

N.B. overlapping broad resonances of same J interference effects

reaction rate through: broad resonances

summary

stellar reaction rate of nuclear reaction determined by the sum of contributions due to

direct transitions to the various bound states

all narrow resonances in the relevant energy window

broad resonances (tails) e.g. from higher lying resonances

any interference term

Rolfs & Rodney

Cauldrons in the Cosmos, 1988

erference

ii

vvvvv inttailsRiDCi total rate