Behavior of Nuclear Reaction Networks - Center for … · Behavior of Nuclear Reaction Networks...
Transcript of Behavior of Nuclear Reaction Networks - Center for … · Behavior of Nuclear Reaction Networks...
Answer to Binding Energy Problem
nMeV/nucleo8.6425256/981.483/)(
483.981-53.9010)(071.8*)2856(289.7*28)56,28()(
nMeV/nucleo8.7943762/251.545/)(
251.545-66.7450)(071.8*)2862(289.7*28)62,28()(
62
56
62
62
==
=−−+==
==
=−−+==
ANiB
MeVBNiB
ANiB
MeVBNiB
NSE Calculator
http://nucleo.ces.clemson.edu/pages/nse/0.1
Detour (a simple minimization problem)
• Minimize the function f(x,y)=x2+y2
0,0
,
022
==
=+=
yx
so
dydxarbitrary
ydyxdxdf
A more interesting example• Minimize the function f(x,y)=x2+y2 subject to the
constraint that y=-x+b, b a constant
2/
0)22(22
022
bybyyso
yxdxyxydxxdxdf
dxdy
nowbutydyxdxdf
=⇒+−=
=⇒=−=−=
−=
=+=
A better way: Lagrange multipliers• Minimize the function f(x,y)=x2+y2 subject to the
constraint that y=-x+b, b a constant
2/2/2/
2/,2/
0)2()2(0)(
,22
byxbbbxy
yxso
dyydxxgfd
Findbxygyxf
Define
==⇒=⇒+−=⇒+−=
==
=−+−⇒=−
−+=+=
λλλλλ
λλλ
nipii
npiniiiii
npp
nn
NZ
ZAZAOthers
otons
Neutrons
µµµ
µµµµλλµ
µµλλλµ
µλλµ
+=⇒
−+=⇒=−−
−=⇒=−−
=⇒=−
)(0:
0:Pr0:
21
221
11
Chemical potentials
• Consider nuclei as Maxwell-Boltzmannparticles:
+=
2/322
),(2),(),(ln),(),(
kTAZmh
AZGAZYNkTcAZmAZ A
πρµ
The Nuclear Data Tool
http://nucleo.ces.clemson.edu/home/online_tools/nuclear_data/0.1
(n,gamma)-(gamma,n) equilibrium
• Nuclei in equilibrium under exchange of neutrons
• Beta decay:
ZZZZZ
AZ
AZZ
YYdtdY
then
and
AZYYwhere
AZYAZY
λλ
λλ
−=
=
=
−−
∑
∑
11
),(
),(),(
Tasks for today• NSE
– Compute the NSE abundances from the expressions in slides 13 and 14
– Determine at what time record in alpha-rich.fits (fits file from yesterday) that the network abundances fall out of NSE
• QSE– Confirm the expressions in slide 17 by minimizing the
free energy f subject to the constraints in slide 16– Confirm that the network abundances in alpha-rich.fits
satisfy these relations after they fall out of NSE