Nuclear Reactions in Stars & in the Laboratory

Nuclear Reactions in Stars & in the Laboratory
Experimental techniques and formalism to simulate
stellar burning processes in the laboratory and derive
stellar reaction rates for nuclear reaction processes.

Reaction Rate Definition
For a given relative velocity v with projectile number density n p
λ
=
σ
⋅
n
p
⋅
v
[ ] s
−1
reaction/target particle
energy/temperature dependent decay constant λ
R
=
σ
⋅
n
p
⋅
v
⋅
n
T
⋅V
[ ] s
−1
reaction rate in volume V

Reaction Rate in Stellar Environment
reaction rate per second and cm 3 :
r
= n ⋅n
⋅σ
⋅
p
T
v
Reaction rate for particles with velocity distribution Φ(v)
r
1
= n ⋅ n ⋅∫σ
⋅v
⋅Φ(
v)
⋅ dv
p T
1+
δ
pT
Accounting for reactions
Between identical particles

Maxwell Boltzmann Distribution
In stellar material of temperature T particles follow ideal gas law
Φ(
v)
4
=
3/
2
2
m v
⎛ m ⎞ −
2
⎜ ⎟
∫
2kT
π v e ( v dv =
with
4
⎜
2 kT
⎟
⎝ π ⎠
Φ ) 1
arbitrary units
3
2
1
max at
E=kT
example: in terms
of energy
E=1/2 m v 2
0
0 20 40 60 80
energy (keV)

Temperature in Stars

Stellar reaction rates
r
1
= YT
Ypρ
2 2
NA
< σv
1+
δ
pT
>
reactions per s and cm 3
1
λ = Y p
ρ NA
< σv
1+
δ
pT
>
reactions per s &
Target nucleus
this is usually referred to
as the stellar reaction rate
units of stellar reaction rate N A
: usually cm 3 /s/mole
n
T
X
T
= ρ ⋅ N
A
⋅ = ρ ⋅
AT
N
A
⋅Y
T
X T ; mass fraction
Y T : abundance

Gamow-Range & Reaction Rate
Stellar Energy Range -- Gamow Window
-- Resonance Width
∝ exp ( - E / kT )
GAMOW PEAK
σ ∝ exp ( - b / √E )
∝ exp ( - E / kT )
RESONANCE
σ ∝
Γ 2
( E - E ) 2 + (Γ/ 2) 2
Nonresonant Reaction Contributions
N < σ v > ∝ T
A
-3/2
∫
σ E exp ( - E / kT ) d E
σ: cross section
N
Resonant Reaction Rate
-3/2
A < σv > ∝ T ωγ
exp ( - E / kT )
R
ωγ: res. strength
E R : res. energy

The Gamow Range of Stellar Burning
The Gamow window or the range of relevant cross section
for “non-resonant” processes is calculated:
Check derivation in book
3/ 2
⎛ bkT ⎞
E0 = ⎜ ⎟ = 0.122⋅
1 2
T
⎝ 2 ⎠
(
2 2
)
1/3 2/3
Z Z A MeV
9
ΔE
=
4 6
E0kT
= 0.2368⋅
1 2
T9
3
(
2 2
)
1/ 6 5/
Z Z A MeV
with A “reduced mass number” and T 9
the temperature in GK

The Gamow peak for 12 C(p,γ) 13
13 N
Note:
kT=2.5 keV !

Examples of Gamow window energies
EG amow [M eV]
10.00
1.00
0.10
0.01
0.0 0.1 1.0 10.0
temperature [GK]
p+p
12C+p
12C+a
12C+12C
strong dependence
on Z & temperature

Change in Abundance
A + a ⇒ B
A reaction is a random process with a reaction probability
(reaction rate) and follows the laws of radioactive decay:
Depletion of isotope A
Formation of isotope B
dn
dt
dn
dt
A
B
= −n
λ = −n
Y ρ NA
= + n
A
A
λ
A
a
< σ v
>
consequently:
n
A
( t)
=
n
0 A
e
−λ
t
n
B
( t)
=
n
0 A
(1 −
e
−λ
t
)

0.007
Stellar lifetime of nuclei
abundance
0.006
0.005
0.004
0.003
0.002
0.001
Y A (t)
τ
Y B
(t)
same
abundance
level Y A
(t=0)
0
10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 5
time
Y
Y
A
B
( t)
( t)
=
=
Y
Y
0 A
0 A
e
−λ
t
(1 −
e
−λ
t
)
τ =
1
=
λ
1
Y a
ρ < σ v
N A
>

Energy production
Reaction Q-value: Q
Energy generated (if Q>0) by a single reaction
Q
⎛
2
= c ⎜
∑ m − ∑
i
m j
⎝ initial nuclei i final nuclei j
⎞
⎟
⎠
Difference between masses in entrance and exit channel
Energy generation: Energy generated per g and sec by a reaction:
ε
r ⋅Q
1
= = Q ⋅ Y ⋅Y
⋅ ρ ⋅ N
2
< σ v
A a
ρ 1+
δ
A
aA
>

Reaction Flow
Reaction flow is defined as the net # of nuclei converted
in time T from species A to B via a specific reaction
F
=
T
dY
⎛ ⎞
∫ ⎜
A
⎟ dt = ∫
⎝ dt ⎠
0
via specific reaction
T
0
λ( t)
Y
A
( t)
dt
Reaction path is usually defined as the sequence of reactions
with maximum reaction flow in a certain stellar environment!

Reaction Network Simulations
Reaction Network Simulations
Change of istopic abundances:
d Y
d t i
+
i
j,k,l
of Stellar Nucleosynthesis
= Σ N λ Y + N ρ N < j,k >
j
2 2
A j k l
Σ N ρ N < j,k,l > Y Y Y
j,k,l
i
j
j
(p,γ)
(γ,p)
j
Σ j,k
+
(β ν)
i
j,k A j k
(α,γ)
Reaction flux = net number of reactions:
Y Y +
Coupled system of nonlinear
differential equations

A+p
capture
Resonance
Cross-Section and S
Cross section and reaction rate
Section and S-Factor
B
energy
cross section often expressed as S-factor
A+p
A+p
Resonance
B
direct
capture
B
cross section
S(E) = σ(E) E exp(2πη)
direct
capture
uncertainties at low energies
due to deviations from
Resonance
l=0 Coulomb penetrability
and due to threshold contributions
energy
cross section often expressed as S-factor
S-factor
direct
capture
energy
Resonance
S-factor to correct for Coulomb barrier: S(E) = σ(E) E e 2πη
S(E) = σ(E) E exp(2πη)
-factor
direct
capture

Example: 12 C(p,γ) cross section
need cross section
here !
2π
⋅η
=
2π
⋅ Z
h
2
1
⋅ Z
2⋅
E
μ
2
2
cm
⋅e
2
S(E) = σ ⋅E
⋅e
= 31.38⋅
Z
2
1
E
⋅ Z
cm
2π
⋅η
2
2
⋅
μ
[ MeV ]

S-factor Conversion
S≈1.3·10 -3 MeV-b
From the NACRE compilation of charged particle induced reaction rates on
stable nuclei from H to Si (see C. Angulo et al. Nucl. Phys. A 656, 3 (1999);
or: http://pntpm3.ulb.ac.be/Nacre/barre_database.htm)

Reaction rate from S-factorS
If S-factor ~ constant over the Gamow range
the rate is calculated in terms of the S-factor
S(E)=S(E 0 )
N
A
< σv
>=
7.83⋅10
9
⎛
⎜
⎝
Z1Z
μT
9
2
⎞
⎟
⎠
1/3
S(
E
0
)[MeV barn]
e
⎛
−4.2487⎜
⎝
2
1
Z Z
T
9
2
2
μ ⎞
⎟
⎠
1/ 3
Otherwise energy dependence needs to be approximated!

Example: 12 C(p,γ) 13
13
N
Calculate the reaction rate as function of temperature!

Calculate life time of 12 C
Determine 12 C life time in the sun T=0.015 GK!
τ≈6 10 13 s = 2 10 6 y

Resonant Reaction Rate
N
A
σv
=
.54⋅10
⋅ωγ
[ MeV ]
⎛ 1
⋅
⎜
⎝ μ ⋅T
⎞
⎟
⎠
3/ 2
⋅e
⎛ 11.605⋅E
−
⎜
⎝ T
R
1
11
9
9
[ MeV ]
⎞
⎟
⎠
ωγ =
2
( J + 1)
Γ
⋅Γ
in out
⋅
Γ =
( ) ( )
∑ tot
Γ
i i
j + 1 ⋅2
j + 1 Γ
p
2
T
tot
Γ
in
Θ
p,
α
at low energy astrophysical conditions: Γ in

12 C(
13 N
Example: Calculate the 12 C(p, p,γ) 13
resonance rate
Resonance strength:
Resonance energy:
ωγ= 0.65 eV= 6.5 10 -7 MeV
Experimental value: E
cm
r =0.45 MeV
Assumed value: E
cm
r =0.05 MeV
Significantly higher
impact at lower energies!

Comparison between resonant and
non-resonant contributions!
Low energy resonances limit the
impact of non-resonant components!

Impact of effective life-time of isotope
Fictitious low energy resonance reduces the life-time
of solar 12 C from ~2 Million years to ~1 year only !

Reaction rates for inverse Processes
Detailed balance between
forward & inverse reaction.
A(a,b)B
The reaction rate ratio
between the forward &
the inverse reaction
depends on the Q-value
and the temperature T!
kT=T[GK]/11.605
μ=m 1·m 2
/(m 1
+m 2
)

Photodisintegrations Processes
Equilibrium between forward
and inverse processes
Saha equation
In units [1/s]