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

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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 heavier“target”

nucleus(Lab: target)

the lighter“incoming projectile”(Lab: beam)

the lighter “outgoingparticle”(Lab: residualof beam)

the heavier

“residual”nucleus(Lab: residual of target)

(adapted from traditional laboratory experiments with a target and a beam)


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12C(p,g)13N

(b+)13C

(p,g)14N

(p,g)15O

(b+)15N

(p,a)12C

Need reaction rates to:

  • establish existence of cycle

  • determine total rate of cycle

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 -> a + 2e+ + 2ve

But requires C or N as catalysts (second and later generation star)


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Sun

pp reaction chain:



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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

with j as particle number current density.Of course j = n v withparticle number density n)

l = s j

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 nucleiI=jA : beam number current (number of particles per second hitting the target)

note: dnTis 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 v+dv is

with

example: in termsof energyE=1/2 m v2

max atE=kT


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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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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 massm 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 NA<sv>: 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 -> 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 A+a) :

(of course half-life of A T1/2=ln2/l)


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5. 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:

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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Basic reaction mechanisms involving strong or electromagnetic interaction:

Example: neutron capture A + n -> B + g

I. Direct reactions (for example, direct capture)

direct transition into bound states

g

En

Sn

A+n

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

A+n

B

B


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or a resonant A(n, electromagnetic interaction:a)B reaction:

Step 1: Compound nucleus formation (in an unbound state)

Step 2: Compound nucleus decay

a

En

G

Sn

A+n

Sa

C

B+a

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 …