9. Direct reactions - for example direct capture:

1
9. Direct reactions - for example direct capture
a A -gt B g
Direct transition from initial state aAgt to
final state ltf (some state in B)
geometrical factor(deBroglie wave length of
projectile - size of projectile)
Penetrability probabilityfor projectile to
reach the target nucleus forinteraction.
Depends on projectileAngular momentum land
Energy E
Interaction matrixelement
III.25
2
Penetrability
2 effects that can strongly reduce penetrability
1. Coulomb barrier
V
for a projectile with Z2 anda nucleus with Z1
Coulomb Barrier Vc
Potential
R
r
or
Example 12C(p,g) VC 3 MeV
Typical particle energies in astrophysics are
kT1-100 keV !
Therefore, all charged particle reaction rates in
nuclear astrophysics occur way below the
Coulomb barrier fusion is only possible
through tunneling
3
2. Angular momentum barrier
Incident particles can have orbital angular
momentum L
Momentum pImpact parameter d
Classical
p
d
In quantum mechanics the angular momentum of an
incident particle can havediscrete values
With
s-wave
And parity of thewave function (-1)l
l 0
p-wave
l 1
l 2
d-wave

For radial motion (with respect to the center of
the nucleus), angular momentumconservation
(central potential !) leads to an energy barrier
for non zero angularmomentum.
Classically, if considering the radial component
of the motion, d is decreasing, which requires an
increase in momentum p and therefore in energy to
conserve L.
4
Energy E of a particle with angular momentum L
(still classical)
Similar here in quantum mechanics
m reduced mass of projectile-target system
Peaks again at nuclear radius (like Coulomb
barrier)
Or in MeV using the nuclear radius and mass
numbers of projectile A1 andtarget A2
5
9.1. Direct reactions the simplest case s-wave
neutron capture
No Coulomb or angular momentum barriers
Vl0
VC0
s-wave capture therefore always dominates at low
energies
But, change in potential still causes reflection
even without a barrierRecall basic quantum
mechanics
Incoming wave
transmitted wave
Reflected wave
Potential
Transmission proportional to
6
Therefore, for direct s-wave neutron capture
Penetrability
Cross section (use Eq. III.19)
Or
Example 7Li(n,g)
thermalcross sectionltsgt45.4 mb(see Pg. 27)
1/v
Deviationfrom 1/vdue to resonantcontribution
7
Why s-wave dominated ? Level scheme
2.063
3/2- 1/2
E1 g
7Li n
0.981
1
E1 g
0
2
8Li
Angular momentum and parity conservation
l
Entrance channel 7Li n
3/2- 1/2 l(-1) 1-, 2-
( l0 for s-wave )
Exit channel 8Li g
2 ? (photon spin/parity)
Recall Photon angular momentum/parity depend on
multiploarity
For angular momentum L (multipolarity) electric
transition EL parity (-1)L

magnetic transition
ML parity (-1)L1
8

• Also recall
• E.M. Transition strength increases
• for lower L
• for E over M
• for higher energy

Entrance channel 7Li n
3/2- 1/2 l(-1) 1-, 2-
( l0 for s-wave )
Exit channel 8Li g
2 1-
1-, 2-, 3-
E1 photonlowest EL that allows to
fulfillconservation laws
match possible
Same for 1 state
At low energies 7Li(n,g) is dominated by
(direct) s-wave E1 capture.
9
Stellar reaction rate for s-wave neutron capture
Because
Thermal neutron cross section
Many neutron capture cross sections have been
measured at reactors usinga thermal (room
temperature) neutron energy distribution atT
293.6 K (20 0C), kT25.3 meV
The measured cross section is an average over the
neutron flux spectrum F(E) used
(all Lab energies)
For a thermal spectrum
so
10
?
Why is a flux of thermalized particles
distributed as
The number density n of particles in the beam is
Maxwell Boltzmann distributed
BUT the flux is the number of neutrons hitting
the target per second and area. Thisis a current
density j n v
therefore
The cross section is averaged over the neutron
flux (the number of neutronshitting the target
for each energy bin) because that is what
determines the event rate.
This is the same situation in the center of a
star. The number density of particlesis M.B.
distributed, but the number of particles passing
through an area per secondis
distributed, and so is the stellar reaction rate !
11
With these definitions one can show that the
measured averaged cross sectionand the stellar
reaction rate are related simply by
(most frequent velocity, corresponding to ECMkT)
with
for reactor neutrons (thermal neutrons)
vT2.20e5 cm/s
thats usually tabulated asthermal cross
section
and
For s-wave neutron capture (which is generally
the only capture mechanism for room temperature
neutrons) one can relate the thermal cross
section to the actual cross section value at the
energy kT
which is for example useful to read cross section
graphs
12
9.2. Direct reactions neutron captures with
higher orbital angular momentum
For neutron capture, the only barrier is the
angular momentum barrier
The penetrability scales with
and therefore the cross section (Eq III.19)
for lgt0 cross section decreases with decreasing
energy (as there is a barrier present) Therefore,
s-wave capture in general dominates at low
energies, in particular at thermal energies.
Higher l-capture usually plays only a role at
higher energies. What higher energies means
depends on case to case - sometimes s-wave is
strongly suppressed because of angular momentum
selection rules (as it wouldthen require higher
gamma-ray multipolarities)
13
Example p-wave capture in 14C(n,g)15C
(from Wiescher et al. ApJ 363 (1990) 340)
14
Why p-wave ?
14Cn
0.74
5/2
1/2
0
15C
Exit channel (15C g)
g
total to 1/2
total to 5/2
E1
1/2- 3/2-
3/2- 5/2- 7/2-
strongest !
1-
1
3/2 5/2 7/2
M1
1/2 3/2
E2
1/2 3/2 5/2 7/2 9/2
2
3/2 5/2
Entrance channel
strongest possible Exit multipole
n
total
into 1/2
into 5/2
lp
14C
M1
E2
s-wave
0
1/2
0
1/2
3/2-
p-wave
1/2-
E1
1-
0
1/2
E1
despite of higher barrier, for relevant energies
(1-100 keV) p-wave E1 dominates.At low energies,
for example thermal neutrons, s-wave still
dominates. But herefor example, the thermal
cross section is exceptionally low (lt1mb limit
known)
15
9.2.1. What is the energy range the cross section
needs to be known to determine the
stellar reaction rate for n-capture ?
This depends on cross section shape and
temperature
s-wave n-capture
of the order of KT (somewhat lower than MB
distribution)
Example kT10 keV
M.B. distribution Y(E)
s(E) Y(E) E E1/2 exp(-E/kT) relevant for
stellar reaction rate
s(E)
16
p-wave n-capture
of the order of KT (close to MB distribution)
Example kT10 keV
M.B. distribution Y(E)
s(E) Y(E) E E3/2exp(-E/kT)relevant for stellar
reaction rate
s(E)
17
9.2.2 The concept of the astrophysical S-factor
(for n-capture)
recall
III.25
trivial strongenergydependence
real nuclear physicsweak energy
dependence(for direct reactions !)
S-factor concept write cross section as
strong trivial energy dependence X
weakly energy dependent S-factor

• The S-factor can be
• easier graphed
• easier fitted and tabulated
• easier extrapolated
• and contains all the essential nuclear physics

Note There is no universally defined S-factor -
the S-factor definition depends on
Pl(E) and therefore on the type of reaction and
the dominant l-value !!!
18
For neutron capture with strong s-wave dominance
with corrections. Then define S-factor S(E)
and expand S(E) around E0 as powers of
with
denoting
in practice, these are tabulated fitted parameters
typical S(E) units with this definition barn
MeV1/2
19
and for the astrophysical reaction rate (after
integrating over the M.B. distribution)
of course for pure s-wave capture
for pure s-wave capture the S-factor is entirely
determined by the thermal cross section measured
with room temperature reactor neutrons
using
one finds (see Pg. 35)
20
For neutron capture that is dominated by p-wave,
such as 14C(p,g) one can definea p-wave S-factor
or
which leads to a relatively constantS-factor
because of
(typical unit for S(E) is then barn/MeV1/2)
S-factor
21
9.3. Charged particle induced direct reactions
9.3.1Cross section and S-factor definition
(for example proton capture - such as 12C(p,g) in
CN cycle)
incoming projectile Z1 A1 (for example proton or
a particle) target nucleus Z2 A2
again
but now incoming particle has to overcome Coulomb
barrier. Therefore
with
(from basic quantum mechanical barrier
transmission coefficient)
22
so the main energy dependence of the cross
section (for direct reactions !)is given by
therefore the S-factor for charged particle
reactions is defined via
typical unit for S(E) keV barn
So far this all assumed s-wave capture. However,
the additional angular momentumbarrier leads
only to a roughly constant addition to this
S-factor that strongly decreaseswith l
Therefore, the S-factor for charged particle
reactions is defined independentlyof the orbital
angular momentum
23
Given here is the partial proton width
Rnuclear radiusQ reduced width (matrix
element)
(from Rolfs Rodney)
24
Example12C(p,g) cross section
need cross sectionhere !
25
S-Factor
Need rateabout here
From the NACRE compilation of charged particle
induced reaction rates on stable nuclei from H
to Si (Angulo et al. Nucl. Phys. A 656 (1999) 3
26
9.3.2. Relevant cross section - Gamov Window
for charged particle reactions
Gamov Peak
Note relevant cross sectionin tail of M.B.
distribution, much larger thankT (very
different from n-capture !)
27
The Gamov peak can be approximated with a Gaussian
centered at energy E0 and with 1/e width DE
Then, the Gamov window or the range of relevant
cross section can be easily calculated using
with A reduced mass number and T9 the
temperature in GK
28
Example
NotekT2.5 keV !
29
9.3.3. Reaction rate from S-factor
Often (for example with theoretical reaction
rates) one approximates the rate calculation by
assuming the S-factor is constant over the Gamov
WIndow
S(E)S(E0)
then one finds the useful equation
Equation III.53
(A reduced mass number !)
30
better (and this is often done for experimental
data) one expands S(E) around E0as powers of E
to second order
If one integrates this over the Gamov window, one
finds that one can useEquation III.46 when
replacing S(E0) with the effective S-factor Seff
and E0 as location of the Gamov Window (see Pg.
51)
with