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Behavior of Nuclear Reaction NetworksBrad Meyer
Clemson University
For the 238U decay chain
238238238238
238238238
238238234234238238234234
238238234234234
1
1
YYt
Ydt
dYand
YYYYt
YYdt
dY
λλ
λ
λλλλ
λλ
−=Δ⎟⎠⎞
⎜⎝⎛ +Δ
−=
+−=Δ−Δ⎟⎠⎞
⎜⎝⎛ +Δ
+−=
Answer to Problem 2.1
⎥⎦
⎤⎢⎣
⎡Δ+
+−+
Δ
=Δ
+Δ
−=Δ
tYY
t
Y
and
t
YY
238
238238234234
234
234
238
238238238
111
1
λλλ
λ
λ
λ
)()(
1)(
,/1
/1
238234
238234234234
238
238238238238
238
234
ttYYYttY
and
tYYYttY
then
tbut
tConsider
Δ+=Δ+=Δ+
Δ+=Δ+=Δ+
<Δ
>>Δ
λλ
λ
λ
λ
S Process
Neutron Sources
• 13C(alpha,n)16O– 13C made from protons diffusing into 12C-rich
region• 22Ne(alpha,n)25Mg
– 22Ne made from 14N left over from CNO:– 14N(alpha,gamma)18F(beta+)18O(alpha,gamma)22Ne
Sites of the s process
• Weak component: core helium burning in massive stars (production up to A=90)
• Main component: helium shell flashes in Asymptotic Giant Branch (AGB) stars (production up to A=209)
Steady states in s process
73
72
72
73
7272737373 0
><><
=⇒
⇒><+><−=
vv
YY
YYvNYYvNdt
dY
Ge
Ge
nGeAnGeAGe
σσ
ρσρσ
A simple helium burning + s-processexample
• Start with abundances after H burning + some extra 13C.
• T9=0.15, density = 1000 g/cc
• The FITS file:
http://www.ces.clemson.edu/~mbradle/JINA/
Choose sprocess.fits. Notice also the reactions file sprocess_reactions.html
The steady state
><=⇒
+−=
vNYYY
YYYvNdt
dY
Aeq
d
eqn
eqp
dnpAp
σλ
λρσ
γ
γ
Consider now equilibrium:p+n<->d
More formally
∑
∑
=⇒
==
+−−=
iii
iii
dYdf
dVdT
Consider
dYPdVsdTdf
μ
μ
0
For p,n, and d network
pdnp
npd
ddnnpp
dYdf
so
dYdYdY
but
dYdYdYdf
)( μμμ
μμμ
−+−=
−=−=
++=
For equilibrium
dnp
so
df
μμμ =+
= 0
General Nucleosynthesis Network
np ZAZAZ μμμ )(),( −+=
Nuclear Binding Energy
AAZBAB
and
AZZAZAZB
so
cAZMcmZAcZmAZB
MevAZAcAZM
np
np
/),(/
),()(),(
),()(),(
),(478.931),(
222
2
=
Δ−Δ−+Δ=
−−+=
Δ+=
The Nuclear Data Tool
http://nucleo.ces.clemson.edu/home/online_tools/nuclear_data/0.1
Ye
∑=i
iie YZY
The NSE Calculator
http://nucleo.ces.clemson.edu/pages/nse/0.1
An alpha-rich freezeout example
• Start with Ye=0.5 at T9=10 and density = 108 g/cc
• Volume expands on a 0.1 second timescale, density proportional to T9
3
• The FITS file:
http://www.ces.clemson.edu/~mbradle/JINA/
Choose alpha-rich.fits
Tasks for today• S process (sprocess.fits)
– Plot the 4He abundance vs. time (in years: 1 year = 3.15e7 seconds)
– Plot neutron number density vs. time – Plot abundances vs. A for several time records– Investigate steady states
• Nuclear Data– Determine the nucleus with the highest B/A (hint: it is between
Z=20 and Z=30)– Determine the nucleus with the highest B/A but with Z=N (It also
lies between Z=20 and Z=30. Is that nucleus stable against beta decay?)
Tasks for Today (cont.)
• NSE Calculator– Compute NSE for fixed T and varying density
or varying T and fixed density. Plot abundances vs. the varying quantity. Try this similarly for varying Ye.
• Alpha-rich freezeout (alpha-rich.fits)– Compare network abundances and NSE
abundances at various time records in the calculation.
• Investigate the Oracle (next)
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