Transcript Introduction to stellar reaction rates Nuclear reactions • generate energy • create new isotopes and elements Notation for stellar rates: p 12C 13N g 12C(p,g)13N The heavier “target” nucleus (Lab: target) the lighter “incoming projectile” (Lab: beam) the.

Introduction to stellar reaction rates
Nuclear reactions
• generate energy
• create new isotopes and elements
Notation for stellar rates:
p
12C
13N
g
12C(p,g)13N
The heavier
“target”
nucleus
(Lab: target)
the lighter
“incoming
projectile”
(Lab: beam)
the lighter
“outgoing
particle”
(Lab: residual
of beam)
the heavier
“residual”
nucleus
(Lab: residual of target)
(adapted from traditional laboratory experiments with a target and a beam)
Typical reactions in nuclear astrophysics:
(p,g)
(p,a)
(p,n)
(n,g)
(n,a)
:
A(B,g)
A(B,p)
A(B,a)
A(B,n)
and their inverses
cross section s
bombard target nuclei with projectiles:
relative velocity v
Definition of cross section:
# of reactions
per second and target nucleus
or in symbols:
l=sj
=
s
.
# of incoming projectiles
per second and cm2
with j as particle number current density.
Of course j = n v with particle number density n)
Units for cross section:
1 barn = 10-24 cm2 ( = 100 fm2 or about half the size (cross sectional area) of a
uranium nucleus)
Reaction rate in stellar environment
Mix of (fully ionized) projectiles and target nuclei at a temperature T
Reaction rate for relative velocity v
in volume V with projectile number density np
l  s n pv
R  s n p vnT V
Reactions per second
so for reaction rate per second and cm3:
r  np nTs v
This is proportional to the number of p-T pairs in the volume.
If projectile and target are identical, one has to divide by 2 to avoid double counting
r
1
1   pT
n p nT s v
Relative velocities in stars: Maxwell Boltzmann distribution
for most practical applications (for example in stars) projectile and target nuclei
are always in thermal equilibrium and follow a Maxwell-Bolzmann velocity
distribution:
then the probability F(v) to find a particle with a velocity between v and v+dv is
 m 

F (v)  4 
 2 kT 
3/ 2
v2 e
m v2

2 kT
with
 F (v)dv  1
4
example: in terms
of energy
E=1/2 m v2
arbitrary units
3
max at
E=kT
2
1
0
0
20
40
energy (keV)
60
80
one can show (Clayton Pg 294-295) that the relative velocities between two particles
are distributed the same way:
  

F (v)  4 
 2 kT 
3/ 2
2
v e

 v2
2 kT
with the mass m replaced by the reduced mass  of the 2 particle system

m1m2
m1  m 2
the stellar reaction rate has to be averaged over the distribution F(v)
r
or short hand:
1
1   pT
r
n p nT  s (v)F(v)vdv
1
1   pT
typical strong
velocity dependence !
n p nT  sv 
expressed in terms abundances
r
l
1
1   pT
1
1   pT
YT Yp  2 N 2A  sv 
Yp  N A  sv 
reactions per s and cm3
reactions per s and target
nucleus
this is usually referred to
as the stellar reaction rate
of a specific reaction
units of stellar reaction rate NA<sv>: usually cm3/s/mole, though in fact
cm3/s/g would be better (and is needed to verify dimensions of equations)
(Y does not have a unit)
Abundance changes, lifetimes
Lets assume the only reaction that involves nuclei A and B is destruction
(production) of A (B) by A capturing the projectile a:
A + a -> B
Again the reaction is a random process with const probability (as long as the
conditions are unchanged) and therefore governed by the same laws as
radioactive decay:
dnA
  n Al   n AYa  N A  s v 
dt
dnB
  n Al
dt
consequently:
n A (t )  n0 A e  l t
nB (t )  n0 A (1  e l t )
and of course
YA (t )  Y0 A e  l t
after some time, nucleus A
is entirely converted to nucleus B
YB (t )  Y0 A (1  e  l t )
Example:
0.007
abundance
0.006
0.005
Y0A
A
B
0.004
0.003
same
abundance
level Y0A
0.002
Y0A/e
0.001
0
-2
10
time
-1
10
0
10
1
10

2
Lifetime of A (against destruction via the reaction A+a) :
(of course half-life of A T1/2=ln2/l)
3
10
10

1
l

4
10
5
10
1
Ya  N A  s v 
Energy generation through a specific reaction:
Again, consider the reaction A+a->B
Reaction Q-value: Energy generated (if >0) by a single reaction
in general, for any reaction (sequence) with nuclear masses m:


Q  c   mi   m j 
final nuclei j 
 initialnucleii
2
Energy generation:
Energy generated per g and second by a reaction:

rQ

Q
1
YAYa N 2A  s v 
1   aA
Reaction flow
abundance of nuclei of species A converted in time in time interval [t1,t2] into
species B via a specific reaction AB is called reaction flow
t2
FA B
t2
 dYA 
 
dt   lA B (t )YA (t )dt

dt A B
t1 
t1
For Net reaction flow subtract the flow via the inverse of that specific reaction
Fnet AB  FAB  FBA
(Sometimes the reaction flow is also called reaction flux)
Multiple reactions destroying a nuclide
14O
example: in the CNO cycle, 13N
can either capture a proton or b decay.
(p,g)
13N
each destructive reaction i has a rate li
Total lifetime
the total destruction rate for the nucleus is then
its total lifetime

1
l

l   li
(b+)
13C
i
1
l
i
i
Branching
the reaction flow branching into reaction i, bi is the fraction of destructive flow
through reaction i. (or the fraction of nuclei destroyed via reaction i)
li
bi 
lj
j