III. Nuclear Reaction Rates

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III. Nuclear Reaction Rates

• Nuclear reactions
• generate energy
• create new isotopes and elements

Notation for stellar rates
12C
13N
p
g
12C(p,g)13N
The heaviertarget nucleus(Lab target)
the lighterincoming projectile(Lab beam)
the lighter outgoingparticle(Lab
residualof beam)
the heavier residualnucleus(Lab residual of
target)
(adapted from traditional laboratory experiments
with a target and a beam)
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Example for a sequence of reactions in stars the
CN cycle
Ne(10)
F(9)
O(8)
N(7)
C(6)
3
4
5
6
7
8
9
neutron number
Net effect 4p -gt a 2e 2ve
But requires C or N as catalysts (second and
later generation star)
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Sun pp reaction chain
4
The 4 CNO cycles
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Competition between the p-pchain and the CNO
Cycle
To understand this (and else)
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1. cross section s
bombard target nuclei with projectiles
relative velocity v
Definition of cross section
of reactions
? . of incoming projectiles per
second and target nucleus
per second and cm2
l s j
with j as particle number current density.Of
course j n v with particle number density n)
or in symbols
Units for cross section
1 barn 10-24 cm2 ( 100 fm2 or about half the
size (cross sectional area) of a
uranium nucleus)
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2. Reaction rate in the laboratory
beam of particles
hits target at rest
area A
j,v
thickness d
assume thin target (unattenuated beam intensity
throughout target)
Reaction rate (per target nucleus)
Total reaction rate (reactions per second)
with nT number density of target nuclei
I jA beam number current (number of
particles per second hitting the target)
note dnT is number of target nuclei per cm2.
Often the target thickness is specified
in these terms.
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3. Reaction rate in stellar environment
Mix of (fully ionized) projectiles and target
nuclei at a temperature T
3.1. For a given relative velocity v
in volume V with projectile number density np
so for reaction rate per second and cm3
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
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3.2. at temperature T
for most practical applications (for example in
stars) projectile and target nucleiare 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 vdv is
with
example in termsof energyE1/2 m v2
max atEkT
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Temperature in Stars
The horizontal axis is the velocity in cm/sec and
the vertical axis is proportional to the
probability that a particle in the gas has that
velocity.
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Maxwell-Boltzmann distribution
Boltzmann distribution probability that any one
molecule will be found with energy E If this
distribution is applied to one direction of
velocity for a molecule in an ideal gas, it
becomes Converting this relationship to
probability in terms of speed in three
dimensions
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Details of calculation
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one can show (Clayton Pg 294-295) that the
relative velocities between two particlesare
distributed the same way
with the mass m replaced by the reduced mass m of
the 2 particle system
the stellar reaction rates has to be averaged
over the distribution F(v)
typical strong velocity dependence !
or short hand
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expressed in terms abundances
reactions per s and cm3
reactions per s and targetnucleus
this is usually referred to as the stellar
reaction rateof a specific reaction
units of stellar reaction rate NAltsvgt usually
cm3/s/mole, though in fact cm3/s/g would be
better (and is needed to verify dimensions of
equations)
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4. 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 -gt 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
consequently
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and of course
after some time, nucleus Ais entirely converted
to nucleus B
Example
A
B
Y0A
sameabundancelevel Y0A
abundance
Y0A/e
t
time
Lifetime of A (against destruction via the
reaction Aa)
(of course half-life of A T1/2ln2/l)
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5. Energy generation through a specific reaction
Again, consider the reaction Aa-gtB
Reaction Q-value Energy generated (if gt0) by a
single reaction
in general, for any reaction (sequence) with
nuclear masses m
Energy generation
Energy generated per g and second by a reaction
6. Reaction flow
abundance of nuclei converted in time T from
species A to B via a specific reaction
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7. Multiple reactions destroying a nuclide
14O
example in the CNO cycle, 13N can either
capture a proton or b decay.
(p,g)
13N
(b)
each destructive reaction i has a rate li
13C
7.1. Total lifetime
the total destruction rate for the nucleus is
then
its total lifetime
7.2. Branching
the reaction flow branching into reaction i, bi
is the fraction of destructive flowthrough
reaction i. (or the fraction of nuclei destroyed
via reaction i)
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8. Determining nuclear reaction rates -
Introduction
Needed is the cross section as a function of
energy (velocity)
The stellar reaction rate can then be calculated
by integrating over the Maxwell Boltzmann
distribution.
The cross section depends sensitively on the
reaction mechanism and the properties of the
nuclei involved. It can vary by many (tens)
orders of magnitude
It can either be measured experimentally or
calculated. Both are difficult.
Experiments are complicated by extremely small
cross sections that prevent direct measurements
of the cross sections at the relevant
astrophysical energies (with a some exceptions)
Typical energies for astrophysical reactions are
of the order of kT
Sun T 10 Mio K
Si burning in a massive star T 1 Bio K
There is no nuclear theory that can predict the
relevant properties of nuclei accurately enough.
In practice, a combination of experiments and
theory is needed.
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Nuclear properties that are relevant for reaction
rates
Nucleons in the nucleus can only have discrete
energies. Therefore, the nucleus asa whole can
be excited into discrete energy levels (excited
states)
Excitation energy (MeV)
Spin
Excitationenergy
Parity ( or - )
5.03
3/2-
3rd excited state
5/2
4.45
2nd excited state
2.13
1/2-
1st excited state
0
3/2-
groundstate
0
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Each state is characterized by

• energy (mass)
• spin
• parity
• lifetimes against g,p,n, and a emission

The lifetime is usually given as a width as it
corresponds to a width in the excitation energy
of the state according to Heisenberg
therefore, a lifetime t corresponds to a width G
the lifetime against the individual channels
for g,p,n, and a emission are usually given as
partial widths
Gg, Gp, Gn, and Ga
with
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A Real Example
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Basic reaction mechanisms involving strong or
electromagnetic interaction
Example neutron capture A n -gt B g
I. Direct reactions (for example, direct capture)
direct transition into bound states
g
En
Sn
An
B
II. Resonant reactions (for example, resonant
capture)
Step 1 Coumpound nucleus formation
(in an unbound state)
Step 2 Coumpound nucleus decay
En
G
G
g
Sn
An
B
B
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or a resonant A(n,a)B reaction
Step 1 Compound nucleus formation
(in an unbound state)
Step 2 Compound nucleus decay
a
En
G
Sn
An
Sa
C
Ba
C
B
For resonant reactions, En has to match an
excited state (but all excited states have a
width and there is always some cross section
through tails)
But enhanced cross section for En Ex- Sn
more later