Behavior of Nuclear Reaction
Networks
Brad Meyer
Clemson University
QSE: Quasi-statistical equilibrium
1)∑ AiYi = 1
i
2)∑ Z iYi = Ye
i
3) ∑ Yi = Yh
i , Z i ≥6
Define
g1 = ∑ AiYi − 1
i
and
g 2 = ∑ Z iYi − Ye
i
and
g 3 = ∑ Yi − Yh
i
then
d ( f − λ1 g1 − λ2 g 2 − λ3 g 3 ) = 0
so
∑ (μ
i
i
− λ1 Ai − λ2 Z i − λ3 )dYi = 0
Neutrons :
μ n − λ1 = 0 ⇒ λ1 = μ n
Pr otons :
μ p − λ1 − λ2 = 0 ⇒ λ2 = μ p − μ n
Others :
μi − λ1 Ai − λ2 Z i = 0 ⇒ μi = Ai μ n + Z i ( μ p − μ n )
⇒ μi = Z i μ p + N i μ n
( Z i < 6)
μi − λ1 Ai − λ2 Z i − λ3 = 0 ⇒ μi = Ai μ n + Z i ( μ p − μ n ) + λ3
⇒ μi = Z i μ p + N i μ n + λ3
NSE
QSE
NSE
N
μiQSE − μiNSE = λ3 + Z i ( μ QSE
μ
μ
μ
)
(
)
−
+
−
p
p
i
n
n
so
⎛ Yi
ln⎜⎜ NSE
⎝ Yi
thus
QSE
Ri = e
QSE
⎛
Y
⎞ λ3
p
⎜
⎟⎟ =
+ Z i ln NSE
⎜Y
kT
⎠
⎝ p
λ3 / kT
Zi
Ni
n
Rp R
⎞
⎛ YnQSE
⎟ + N i ln⎜
⎜ Y NSE
⎟
⎝ n
⎠
⎞
⎟⎟
⎠
Re member
d ( f − λ1 g1 − λ2 g 2 − λ3 g 3 ) = 0
⇒ df = λ1dg1 + λ2 dg 2 + λ3 dg 3
⇒ df = μ n ∑ dX i + ( μ p − μ n )dYe + λ3 dYh
i
⇒ df = ( μ p − μ n )dYe + μ h dYh
(n,gamma)-(gamma,n) equilibrium
Define
g1 = ∑ AiYi − 1
i
and
g 2 = ∑ Z iYi − Ye
i
and
g Z = ∑ Y ( Z , A) − YZ
A
then
d ( f − λ1 g1 − λ2 g 2 − ∑ λZ YZ ) = 0
Z
so
∑ (μ
i
i
− λ1 Ai − λ2 Z i )dY − ∑ λZ dYZ = 0
Z
For a given Z
μi − μiNSE = λZ + Z ( μ p − μ pNSE ) + N i ( μ n − μ nNSE )
so
⎛ Yi
ln⎜⎜ NSE
⎝ Yi
thus
Ri = e
⎛ Yp
⎞ λZ
⎟⎟ =
+ Z ln⎜ NSE
⎜Y
kT
⎠
⎝ p
λZ / kT
Z
Ni
n
Rp R
R ( Z , A + 1)
⇒
= Rn
R ( Z , A)
⎞
⎛
⎟ + N i ln⎜ Yn
⎜ Y NSE
⎟
⎝ n
⎠
⎞
⎟⎟
⎠
Parameters
• Ye :
– net number of electrons per nucleon
– fraction of all nucleons that are protons
– E.g., Ye = 0.5 means equal numbers of
neuetron and protons. Ye = 0 means all
neutrons. Ye=1 means all protons.
Parameters (cont.)
• Entropy per nucleon
– Degrees of freedom per nucleon
– In units of Boltzmann’s constant kB
– E.g., all free nucleons: s/kB = about 3
– E.g., all nucleons in 56Fe: s/kB about 3/56
– s/kB > 20 typically means a lot of photons and
electron-positron pairs around.
Parameters (cont.)
• Timescale tau:
– Density expansion efolding timescale
– Typically milliseconds to seconds
The Alpha-Rich Freezeout
Explosive Silicon Burning
The r-Process (I)
BBN
The r-Process (II)
Assembling 4He from n,p
• Early: reaction sequences like
p(n,g)d(n,g)t(d,n)4He
• Later: catalysis by reaction cycles like
56Fe(n,g)57Fe(n,g)58Fe(p,g)59Co(p,4He)56Fe
New FITS files to play with
• R process files: rprocess.html
• Big bang calculations: bbang1.fits (s/k=2x1010),
bbang2.fits (s/k=2x109), bbang3.fits (s/k=2x1011)
Take home message I:
Steady states, steady states, steady
states
Take home message II:
Constrained equilibria,
constrained equilibria, constrained
equilibria